DOI 10.1007/s10958-015-2379-4 Journal of Mathematical Sciences, Vol. 207, No. 3, June, 2015

HOMOGENEOUS SPACES WITH INNER METRIC AND WITH INTEGRABLE INVARIANT DISTRIBUTIONS V. N. Berestovskii and V. V. Gorbatsevich

UDC 514.7, 512.546.3, 517.977

Abstract. This paper is a survey of results on homogeneous spaces of Lie groups G with compact stabilizer subgroup H, on which every G-invariant distribution is integrable. Some unsolved problems are suggested.

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Locally Compact Homogeneous Geodesic Spaces . . . . . . . . . . . . . . . . . . . . . . . Semigroups and Left-Invariant Geodesic Metrics on Locally Compact Topological Groups Homogeneous Geodesic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Nonholonomic Metric Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Search for Geodesics in Homogeneous Manifolds with Inner Metric . . . . . . . . . . Characterization of Invariant Totally Nonholonomic Distributions and Homogeneous Manifolds with Integrable Invariant Distributions . . . . . . . . . . . . . . . . . . . . . . . Strong Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Steps in the Study of Strong Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . Isotropically Irreducible Homogeneous Riemannian Spaces . . . . . . . . . . . . . . . . . . Compact Simply Connected Homogeneous Manifolds with Integrable Invariant Distributions Strong Subalgebras in Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . Structure of Strong Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Subgroups in Solvable Lie Groups and Strict Abelian Subgroups . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

410 412 416 421 427 432 438 441 442 445 448 451 453 457 460

Introduction

Natural generalizations of homogeneous Riemannian and Finsler manifolds are general homogeneous locally compact spaces with inner metric. The naturalness and importance of such generalization can be explained by the following statement. Statement. The simple conditions above are minimal assumptions for fairly meaningful results. The following results are the most important of them. 1. Every homogeneous manifold with inner metric is isometric to a homogeneous (quotient) space of some connected Lie group by a compact (stabilizer) subgroup, supplied with an invariant (maybe, nonholonomic) Finsler metric. 2. The class Ω of all locally compact homogeneous spaces with inner metric admits a metric dBGH , such that (Ω, dBGH ) is a complete separable metric space. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 126, Geometry, 2013.

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c 2015 Springer Science+Business Media, Inc. 1072–3374/15/2073–0410 

3. All subclasses of homogeneous manifolds with inner metric, homogeneous Finsler manifolds, and Lie groups with left-invariant Finsler metric are everywhere dense in (Ω, dBGH ). To justify the above statement, let us note that a (logically scarcely) more general class of complete separable geodesic homogeneous spaces is indeed immense and there are no (and perhaps never will be) meaningful results without highly restricted additional conditions. For example, it is confirmed by the following facts: (a) any two-point homogeneous locally compact space with inner metric is isometric to an Euclidean or rank-one Riemannian symmetric space; (b) any three-point homogeneous locally compact space with inner metric is isometric to a complete homogeneous Riemannian manifold of constant sectional curvature and n-point homogeneous for every finite natural number n; (c) the subclass of infinite-dimensional complete separable geodesic spaces, which are n-point homogeneous for every finite natural number n, includes homogeneous spaces of constant nonpositive sectional curvature, spheres in Hilbert space and corresponding infinite-dimensional elliptic spaces, and the very interesting and not entirely investigated universal metric space of P. S. Urysohn, but nothing more is known about this subclass. More details about this are in the main text of this survey. The above statements and the intensive study of homogeneous Riemannian and Finsler manifolds and their nonholonomic generalizations in the last decades confirm the naturality and importance of investigation of the class Ω. As a corollary of results mentioned above, it is necessary to describe homogeneous spaces of Lie groups by their compact subgroup such that any invariant inner metric on them is a Finsler metric. These are exactly homogeneous spaces with integrable invariant distributions (HSIID), which are the main subject of this survey. The importance of classification problems of the class HSIID is explained by at least two reasons. This class is a sufficiently narrow subclass of the class of all homogeneous manifolds. As a corollary, its complement, consisting of homogeneous manifolds that admit an invariant Finsler, as well as Riemannian metrics (so-called sub-Riemannian or Carnot–Caratheodory metrics), actively investigated three decades after works of M. Gromov, is very broad. On the other hand, the class HSIID includes symmetric Cartan spaces, isotropy irreducible (in particular, strictly isotropy irreducible) homogeneous spaces, classified in simply connected case in the works of M. Wang and W. Ziller (respectively, O. V. Manturov, J. A. Wolf, and M. Kr¨ amer), and also commutative Lie groups and Lie groups such that any left-invariant Riemannian metric on them has a constant negative sectional curvature. Compact simply connected homogeneous spaces from the class HSIID are characterized as direct products of compact simply connected homogeneous strictly isotropy irreducible spaces (see [18]). All spaces G/H enumerated here have the following remarkable properties: (a) if a space does not contain simultaneously compact and noncompact direct factors from the class HSIID, then the Ricci curvature of arbitrary invariant Riemannian metric on the space is zero or of constant sign; (b) for every invariant Riemannian metric μ on G/H, the space (G/H, μ) is isometric to some geodesic orbit space (G1 /H1 , μ1 ), i.e., any geodesic of the space (G1 /H1 , μ1 ) is an orbit of some one-parameter subgroup in G1 . Possibly, both properties (a) and (b) are characteristic for spaces from the class HSIID. In [60], V. V. Gorbatsevich studied in detail general homogeneous spaces with connected stabilizer subgroup from the class HSIID. He described corresponding transitive Lie groups and stabilizer subgroups in the case where transitive group is semisimple or solvable, and also in the case of general 411

transitive Lie groups. These descriptions are related to the notion of strong Lie subalgebra. Also there were given a detailed description of strong Lie subalgebras and their existence condition for a given Lie algebra. Results obtained for spaces from the class Ω are directly related to a survey of nonholonomic geometry presented in Secs. 5 and 6 and methods for the search of geodesics, i.e., locally shortest arcs, on homogeneous manifolds with invariant (including nonholonomic) Finsler and Riemannian metrics. In particular, known results are given that imply that the Hamilton–Jacobi or Euler–Lagrange equations do not permit one in general case to find all geodesics and determine the form of spheres on homogeneous sub-Riemannian manifolds for every (topological) dimension ≥4. In the same sections is discussed the relation of the questions under consideration to geometric group theory, CR-manifolds, screw theory, thermodynamics, etc. The basis of nonholonomic metric geometry is the Rashevsky–Chow theorem, which states that any two points of connected smooth manifolds with totally nonholonomic distributions can be joined by a piecewise smooth path tangent to this distribution. In the particular case of contact distribution, this result has been obtained in the classical 1909 work by C. Caratheodory on axiomatization of thermodynamics, related to the sixth Hilbert problem on mathematical presentation of physics axioms. Left-invariant totally nonholonomic distribution on the Heisenberg group is related to thermodynamics equations, in particular, to equations of the French physicist Sadi Carnot, who published in 1824 the first work on thermodynamics. More details on the works of S. Carnot and C. Caratheodory are given in Secs. 5 and 6. In relation to what has been said, M. Gromov called nonholonomic Riemannian metric Carnot–Caratheodory metric and the corresponding metric spaces Carnot–Caratheodory spaces. V. N. Berestovskii wrote Secs. 1–7 and 11 and Remark 13 of Sec. 12 of this survey. Sections 8 and 12–14 were written by V. V. Gorbatsevich, and Secs. 9 and 10 were written jointly. The general editing of this review was carried out jointly by the authors. Ten unsolved problems are suggested.

2.

Locally Compact Homogeneous Geodesic Spaces

Let us recall the main definitions. A path in a topological space X is a continuous mapping of some closed bounded interval of a real line to the space X. A metric space is called a space with inner metric if the distance between any two of its points is equal to the greatest lower bound of lengths of paths joining these points. A metric space is homogeneous if its motion (isometry) group acts transitively, i.e., for any two points in the space, there is motion of the space moving one of these points to the other. The Cohn-Vossen theorem (see [46]) states that every locally compact complete space (M, ρ) with inner metric is finitely compact, i.e., any closed bounded subset in (M, ρ) (in particular, any closed ball B(x, r) of radius r centered at x) is compact; moreover, the space (M, ρ) is geodesic. The last statement means that any two points of the space can be joined by a segment, i.e., a curve (path) whose length is equal to the distance between these points. Further, we assume that (M, ρ) is an arbitrary locally compact homogeneous space with inner metric ρ, G = I(M ) is its motion group with compact-open topology with respect to its action on (M, ρ), and G0 is the connected component of the unit in the group G. In view of the homogeneity and local compactness, the space (M, ρ) is metrically complete; then the Cohn-Vossen theorem is valid for it, and so we can use the brief term “locally compact homogeneous geodesic space.” On the group G, the Busemann metric is introduced (see [39]): δp (f, g) = sup ρ(f (x), g(x))e−ρ(p,x) , x∈M

412

p ∈ M.

The following results are proved in [11]. The metric δp depends on the choice of the point p ∈ M , but it is bi-Lipshitz equivalent to the metric δq for any point q ∈ M , and thus, independently of the point p ∈ M , defines a topology τ , which coincides with the compact-open topology on G with respect to its action on (M, ρ). Let us recall that the subbasis of the compact-open topology consists of sets G(K; U ) := {g ∈ G | g(K) ⊂ U }, where K and U are some compact and open subsets in M , respectively. The metric δp is invariant under left shifts by elements of the group G and right shifts by elements of the stabilizer H of the group G at the point p (recall that H is a compact subgroup). For the natural identification of G/H with M defined by the formula σ(gH) = g(p), the quotient metric Δp on G/H induced by the metric δp is equivalent to the metric ρ, and the metric space (G, δp ) is locally compact, complete, and separable. The pair (G, τ ) is a topological group acting continuously and properly from the left on (M, ρ) by isometries. Therefore, the subgroup G0 is transitive on M . Moreover, J. Szenthe proved in [122] that M is a manifold and G and G0 are Lie groups if and only if M is locally contractible. The following interesting question is still open. Problem 1 (see [24]). Is it true that in general case the connected group G0 or some other transitive on M closed connected subgroup of the group G is locally connected or, which is equivalent, locally arcwise connected? In connection with this, we note that in [24] is given a short proof of a new (at that time) result, the global form of the theorem on local presentation of a group as a direct product from the Iwasawa– Gleason–Yamabe theory [56, 81, 140] for locally compact groups. Theorem 1. Let G be a connected, locally compact, (Hausdorff ) topological group. Then there exist a compact subgroup K ⊂ G, a connected simply connected Lie group L, and a surjective local isomorphism π : K × L → G. Furthermore, if G is locally connected, then K is connected and locally connected, and π is a covering epimorphism. The following characterization of locally compact homogeneous geodesic spaces as homogeneous spaces of topological groups is valid (see [24]). Theorem 2. Every locally compact homogeneous geodesic space is isometric to some locally compact, locally connected quotient space G/H of a connected, locally compact topological group G with the first countability axiom by a compact subgroup H supplied with a G-invariant geodesic metric. Conversely, every locally connected, locally compact homogeneous space G/H of a connected, locally compact topological group G with the first countability axiom by a compact subgroup H admits a Ginvariant geodesic metric ρ; furthermore, for any number K < 0, one can construct a metric ρ with curvature ≥K in the sense of A. D. Aleksandrov, and a metric ρ with curvature ≥0 if G/H is compact. Note that there are different equivalent definitions of Aleksandrov spaces with the curvature ≥K (see [38]). Here and in [24] is taken the following definition belonging to the first author. Definition 1. A space M with inner metric ρ and local existence of shortest arcs is called an Aleksandrov space with curvature ≥K if locally any quadruple of points from M is isometric to some quadruple of points in a simply connected, complete Riemannian 3-manifold of some constant curvature k ≥ K, where the number k depends on the considered quadruple of points. To formulate other structural results on locally compact homogeneous geodesic spaces, we need the following definition. Definition 2. A mapping of metric spaces f : M → N is said to be an submetry if for any point x ∈ M and any number r > 0, we have f (BM (x, r)) = BN (f (x), r), where B denotes a closed ball of corresponding radius in corresponding space. 413

An example of submetry is the canonical projection π : (G, δp ) → (G/H, Δp ) (see above). Every Riemannian submersion of a complete, smooth Riemannian manifold is a submetry (see [29, 108]); conversely, a submetry of a smooth Riemannian manifold is a Riemannian submersion of class C 1,1 (see [22]). It is not difficult to prove the following statement. Proposition 1. Let (M, ρ) be a locally compact, homogeneous geodesic space, Γ be a compact subgroup in G = I(M ), and M/Γ be the orbit space of the group Γ in (M, ρ). Let us define the distance ρΓ between two orbits O1 , O2 ∈ M/Γ as follows: ρΓ (O1 , O2 ) = min{ρ(x1 , x2 ) : x1 ∈ O1 , x2 ∈ O2 }. Then the canonical projection pr : (M, ρ) → (M/Γ, ρΓ ) is a submetry. Moreover, if Γ is a normal subgroup in G, then (M/Γ, ρΓ ) is locally compact homogeneous geodesic space. Theorem 3 (see [12, 15, 24]). A metric space (M, ρ) is a locally compact, homogeneous geodesic space if and only if it can be represented as an inverse metric limit of some sequence (Mn , ρn ) of homogeneous geodesic manifolds that are related by proper submetries (i.e., submetries such that the preimage of any compact set is compact) pnm : (Mm , ρm ) → (Mn , ρn ), n ≤ m,

pn : (M, ρ) → (Mn , ρn ),

where pn = pnm ◦ pm ,

pns = pnm ◦ pms , n ≤ s ≤ m.

This means that (nonincreasing) functions ρn ◦(pn ×pn ) uniformly converge to the metric ρ. Moreover, pnm ∈ C ∞ , and one can assume that fibers of these submetries are connected. Furthermore, the space (M, ρ) is an Aleksandrov space with curvature ≥K if and only if all manifolds (Mn , ρn ) are homogeneous Riemannian manifolds of sectional curvatures ≥K and the mappings pnm : (Mm , ρm ) → (Mn , ρn ),

n ≤ m,

are Riemannian submersions. The main steps of the proof of this theorem are as follows. Let (M, ρ) be a locally compact homogeneous geodesic space. We may assume that (M, ρ) is not a manifold: otherwise, there is nothing to prove. By the Yamabe theorem (see [140]), any neighborhood U of the unit in a connected topological group G0 contains a closed (even compact) normal subgroup N = NU such that the quotient group G0 /N is a (connected) Lie group. Lemma 3 from the survey [57] states that if N1 and N2 are normal subgroups in a locally compact topological group G such that G/N1 and G/N2 are Lie groups, then G/(N1 ∩ N2 ) is also a Lie group. Choosing an at most countable decreasing base of topology Un of the group G0 at the unit, one can find, based on this lemma, a ˜n ⊂ Un with quotient groups G0 /N ˜n that are Lie decreasing sequence of compact normal subgroups N groups. By [15], the sequence of connected subgroups Nn , which are connected components of the unit ˜n , also possess the properties mentioned above. Using this fact and Proposition 1, we see in the group N that the orbit space Mn = M/Nn supplied with the metric ρn := ρNn is a homogeneous manifold with geodesic metric, and the canonical projection pn : (M, ρ) → (Mn , ρn ) is a submetry. Since Nm ⊂ Nn when n ≤ m, one can naturally define the quotient mapping pnm : Mm = M/Nm → Mn = M/Nn = (M/Nm )/(Nn /Nm ). Moreover, the mapping pnm : (Mm , ρm ) → (Mn , ρn ) is a submetry with connected fibers Nn /Nm (since the group Nn is connected). Now it is easy to prove that the functions ρn ◦ (pn × pn ) uniformly converge to the metric ρ. It is clear that the mapping pnm is C ∞ -smooth with respect to natural smooth structures on the homogeneous manifolds Mm and Mn . If (M, ρ) is an Aleksandrov space 414

with curvature ≥K, then every homogeneous manifold (Mn , ρn ), being the image of the submetry pn : (M, ρ) → (Mn , ρn ), is also an Aleksandrov space with curvature ≥K (see [24]). Then (Mn , ρn ), being also a homogeneous manifold with geodesic metric, is a homogeneous Riemannian manifold with sectional curvatures ≥K. Moreover, every mapping pnm : (Mm , ρm ) → (Mn , ρn ), n ≤ m, being simultaneously a submetry and a C ∞ -mapping of Riemannian manifolds, is a Riemannian submersion of class C ∞ (see [22]). On the other hand, if all spaces (Mn , ρn ) are homogeneous Riemannian manifolds with sectional curvatures ≥ K and all mappings pnm : (Mm , ρm ) → (Mn , ρn ), n ≤ m, are Riemannian submersions with connected fibers, then (Mn , ρn ) are Aleksandrov spaces with curvature ≥K and the space (M, ρ), being a very special case of the Gromov–Hausdorff limit (see Definition 3 below) of the sequence of Aleksandrov spaces (Mn , ρn ) with curvature ≥K, is also an Aleksandrov space with curvature ≥K (see [24]). The converse statement in Theorem 11 is proved by passage to the limit by analogous arguments. In some sense, Theorem 3 reduces the study of locally compact homogeneous geodesic spaces to the case of homogeneous geodesic manifolds, which will be considered in the next section. Recall that the Hausdorff distance dH (A, B) between bounded subsets of an arbitrary metric space M is the greatest lower bound of positive numbers r such that A is contained in an r-neighborhood of the set B and B is contained in an r-neighborhood of the set A. A pair (K(M ), dH ), where K(M ) is the family of all closed bounded subsets of a metric space M , is a metric space; it is complete if the space M is complete (see [89]). Definition 3. The Gromov–Hausdorff distance dGH (A, B) between compact metric spaces is defined as the greatest lower bound of all distances dH (f (A), g(B)) for all metric spaces M and all isometric embeddings f : A → M and g : B → M . By definition, a sequence ((Xn , xn ), ρn ) of finitely compact complete spaces with metrics ρn and chosen points xn converges by Gromov–Hausdorff to a similar space ((X, x), ρ) if for every number r > 0, dGH (BXn (xn , r), BX (x, r)) → 0 as n → +∞. Remark 1. Clearly, in the case of locally compact homogeneous spaces with inner metric of interest to us, the choice of points xn ∈ Xn and x ∈ X does not play any role. In fact, in the definition of dGH , one can take arbitrary fixed metric space M with the following properties: (1) M is complete and separable; (2) M contains some isometric copy of any separable metric space; (3) U is n-point homogeneous for all natural n, i.e., any isometry of two n-point subsets in U can be extended to an isometry of the whole space U onto itself. A metric space with property (2) is said to be universal. For example (see [26]), one can take M = C([0, 1]) with the metric d(f, g) = max{|f (t) − g(t)|, t ∈ [0, 1]} or M = U , where U is the universal Urysohn space (see [125] or the Russian translation of this paper in the collection of main works by P. S. Urysohn [126]). Up to an isometry, the space M = U is characterized by conditions (1)–(3). The space U possesses many remarkable properties; it has been extensively investigated during the last 20 years (see [127, 128] and the references therein). A. M. Vershik showed that one can get the space U as the Hausdorff completion of a countable set supplied with a “random” or general (in the sense of the Baire category) metric. M. Katetov constructed an example of a noncomplete separable metric space with properties (2) and (3). V. Uspensky proved that U is homeomorphic to a Hilbert space, and the topological group I(U ) of all isometries of the space U is universal, i.e., it contains some isomorphic copy of any topological group with a countable base of topology. S. A. Bogatyi proved that every isometry between compact subsets in U can be extended to an isometry of the space U onto itself. Taking M = U in the definition of dGH , one can easily deduce from this that the pair (K, dGH ), where K is the family of all isometry classes of compact metric spaces, is a complete separable metric space. 415

Definition 4. The distance dBGH between finitely compact metric spaces with chosen points (X, x) and (Y, y) is defined as follows:     (2.1) dBGH (X, x), (Y, y) = sup dGH BX (x, r), BY (y, r) e−r . r≥0

By the Cohn-Vossen theorem cited above, this definition is applicable to locally compact complete spaces with inner metric; in particular, to locally compact homogeneous spaces with inner metric. It is clear that in the latter case, the distance dBGH is independent of the choice of points x ∈ X and y ∈ Y . Let Σ and Θ be the classes of all finitely compact metric spaces and all locally compact, complete, inner metric spaces with chosen points, respectively, and Ω be the class of all locally compact homogeneous spaces with inner metric. Using the last statement in Remark 1, one can prove the following theorem. Theorem 4. The pair (Σ, dBGH ) is a complete separable metric space. The convergence of sequences in this metric space is equivalent to their Gromov–Hausdorff convergence. Moreover, Θ and Ω are closed and separable subspaces in (Σ, dBGH ). 3.

Semigroups and Left-Invariant Geodesic Metrics on Locally Compact Topological Groups

Of exceptional interest are questions on the existence, characterization, and structure of leftinvariant geodesic metrics on locally compact (Hausdorff) topological groups. Of course, we assume that the metric and initial topologies on the group coincide. Theorem 5. A locally compact topological group (G, τ ) admits a left-invariant geodesic metric if and only if (G, τ ) is metrizable, connected, and locally connected. It is clear that the conditions of metrizability, connectedness, and local connectedness are necessary. Their sufficiency is a consequence of Theorem 2 for the particular case H = {e} and the Kakutani theorem (see [83]) stating that a topological group is metrizable if and only if it satisfies the first countability axiom (moreover, the group admits a left-invariant metric). The following theorem is a direct consequence of Theorem 3. Theorem 6. A group (G, ρ) is a locally compact topological group with left-invariant geodesic metric if and only if it can be represented as the inverse metric and group limit of a sequence (Gn , ρn ) of connected Lie groups with left-invariant geodesic metric that are related by proper submetries (i.e., submetries for which the preimage of a compact set is compact) and surjective homomorphisms of Lie or topological groups pnm : (Gm , ρm ) → (Gn , ρn ), n ≤ m,

pn : (G, ρ) → (Gn , ρn ),

where pn = pnm ◦ pm , pns = pnm ◦ pms , n ≤ s ≤ m. In particular, the (nondecreasing) functions ρn ◦(pn ×pn ) uniformly converge to the metric ρ. Moreover, pnm ∈ C ∞ , and one can assume that fibers of all these submetries and homomorphisms are connected normal subgroups. Definition 5. The product AB of subsets A and B in any group G is the subset AB = {ab | a ∈ A, b ∈ B}. It is easy to prove the following proposition. Proposition 2. Let (G, τ ) be a locally compact topological group. If ρ is a left-invariant inner (=geodesic) metric on G compatible with the topology τ , then the family of closed (hence compact) balls Br := B(e, r), r ∈ R+ centered at the unit e ∈ G have the following properties: 416

B1.



Br = G;

r≥0

B2. B3. B4. B5.

B0 = {e}; Br−1 = Br ; Br · Bs= Br+s ; Br for all s ≥ 0; Bs = r>s

B6. for every element g ∈ G and every number r > 0, there exists a number s > 0 such that gBs g−1 ⊂ Br . Conversely, if we are given a family Br , r ∈ R+ , of compact subsets in (G, τ ) with conditions B1–B6, then the formula (3.1) ρ(g, h) = inf{r ≥ 0 | g−1 h ∈ Br } defines a left-invariant inner metric on G compatible with the topology τ . Moreover, Br := B(e, r), r ∈ R+ . Properties B1, B2, and B5 are obvious and hold for any metric on G. Properties B3 and B4 were proved in [13, Lemma 7] for the case where G is a Lie group; the first of them is valid for any leftinvariant metric on G, while the second characterizes left-invariant geodesic metrics (for the case of complete metric). Property B6 means that the metric topology defines some structure of topological group on G. Properties B2 and B4 mean that the family Br , r ∈ R+ , constitutes a one-parameter commutative semigroup with unit. Moreover, the group (G, ρ) is compact if and only if for some number D > 0, Br = BD if r ≥ D (one can take the diameter of compact metric space (G, ρ) as D). If on a group G, a semigroup of subsets Br , r ∈ R+ , with properties B1–B4 is given, then formula (3.1) defines a left-invariant metric on G. In formula (3.1), it is possible to change inf by min if and only if condition B5 holds. Moreover, any set of the form Br (g) = lg (Br ), where g ∈ G and r ≥ 0, is a closed ball of radius r centered at g with respect to the metric ρ. If, in addition, the metric space (G, ρ) is complete, then the space (G, ρ) is geodesic and hence arcwise connected and locally arcwise connected. In particular, the space (G, ρ) is complete if it is locally compact. The same is valid if (G, τ ) is a Hausdorff topological group, all sets Br , r ≥ 0, are compact in (G, τ ), and they satisfy conditions B1–B5. In this case, the compactness of the sets Br , r ≥ 0, and condition B5 for r = 0 imply that for any neighborhood U of the point e in (G, τ ), there is a number r > 0 such that Br ⊂ U . As a consequence, the identity mapping idG : (G, ρ) → (G, τ )

(3.2)

is continuous and the group (G, τ ) is arcwise connected. Conditions B1–B6 were first introduced by H. Radstr¨om (see [112]) Definition 6. An (abstract) group G together with a family of subsets Br , r ≥ 0, satisfying conditions B1–B6 from Proposition 5 is said to be normed. Moreover, a function ν : G → R+ , where ν(g) = min{r ≥ 0 | g ∈ Br },

(3.3)

is called a norm on G. A topological group (G, τ ) is aid to be normable if G can be normed in such a way that the topology τ coincides with the metric topology of the metric ρ defined by formula (3.1). Remark 2. A more general notion of geometry on group was introduced and examined in [25]. Note that (3.4) Br = {g ∈ G | ν(g) ≤ r}. Theorem 5 can be reformulated as follows. Theorem 7. A locally compact topological group (G, τ ) is normable if and only if (G, τ ) is metrizable, connected, and locally connected. 417

The necessity of metrizability, connectedness, and separability conditions is marked in [112]. In the same paper the following question is posed: Is it true that every locally compact, separable, metrizable, connected, and locally connected topological group is normable? This problem is solved positively in [112] only for some particular cases: commutative groups, quotients of compact groups by some their totally disconnected closed central subgroups, and Lie groups. Let us note that one can omit the separability condition in Radstr¨om’s question since it is not difficult to prove that any locally compact, metrizable, connected topological group is separable. Therefore, Theorem 7 gives a positive answer to Radstr¨om’s question in the general case. In [112, Sec. 2.8], Radstr¨om discussed in detail some results and problems from A. Gleason’s paper [55]. Let us mention the most important ones. Gleason considered one-parameter semigroups  φ of φ(r), compact sets with the condition φ(0) = {e}. It follows from [55, Lemma 1] that if φ(0) = r>0

then all sets φ(r) are connected. He gives a hint of the proof for the statement that any connected, locally compact, metrizable group either admits nonconstant one-parameter semigroups of such type or contains arbitrary small connected, compact subgroups larger than {e}. He deduced from this result an important consequence on the existence of nontrivial arcs in such groups. Gleason also posed the following problem: Find topological groups G that admit a one-parameter semigroup φ of sets such that the sets φ(r), r > 0, form a fundamental system of neighborhood of e. If a group G is connnected and locally compact, then it is not difficult to prove that sets Br := φ(r), r ≥ 0, are compact and also constitute a one-parameter semigroup satisfying all conditions B1–B6, except for, perhaps, B3. Then formula (3.1) defines a left-invariant geodesic, but which may be a nonsymmetric “metric” on G. This implies the metrizability and the local connectedness of the group G. In view of Theorems 5 and 7 we obtain: A locally compact group is a solution of Gleason’s problem if and only if it is metrizable and locally connected (but is not necessarily connected). The following problem is very interesting. Problem 2. Is it possible to norm a locally compact, metrizable, connected, and locally connected topological group (in particular, a connected Lie group), considered as an abstract group, so that the group with the topology induced by the norm is not isomorphic to the initial topological group? The most important achievements of Radstr¨om connected with the notion of a normed group are his results on general one-parameter semigroups of compact sets on Lie groups and their applications to general left-invariant geodesic metrics on Lie groups, but it is more appropriate to discuss this in detail in the next section. A left-invariant inner metric on a topological group can be characterized not only in terms of one-parameter semigroups of subsets, as in Proposition 2, but also on the language of semigroups of topological groups (see [20]). In order to formulate the corresponding statement, we need two definitions. Definition 7. A (sub)semigroup S of a topological group G, i.e., a set with the condition SS ⊂ S, is said to be locally generated if its intersection with any neighborhood U of the unit e in G topologically generates S, i.e., ∞  (U ∩ S)n . (3.5) S⊂ n=1

Definition 8. A (sub)semigroup S of topological group G is said to be sharp if (a) e ∈ S; 418

(b) for any neighborhood U of the unit, there exists a neighborhood V of the unit such that if x, y ∈ S and xy ∈ V , then x, y ∈ U . Theorem 8 (see [20]). Let (G, τ ) be a complete, metrizable, Hausdorff topological group. The following statements for a nonnegative real-valued continuous function ν on G are equivalent: (a) the epigraph of the function ν E = {(g, t) ∈ G × R : t > ν(g)} is an open, sharp, locally generated subsemigroup of the group G × R, invariant under the transformation i : (g, t) → (g−1 , t); (b) the formula ρ(g, h) = ν(g −1 h) defines a left-invariant inner metric on G compatible with τ . Complete topological groups are defined in [34, p. 52]. Every locally compact group is complete. The statements about left-invariant inner metrics on topological groups given above correspond to dual (in some sense) statements about left-invariant antimetrics (see [20]). A function θ : G × G → [−∞, 0) ∪ (0, +∞) is called an antimetric if θ(f, h) ≥ θ(f, g) + θ(g, h)

for all

f, g, h ∈ G,

and also θ(G × G) ∩ (0, +∞) is not empty. It is easy to see that if θ is a left-invariant antimetric on an arbitrary group G, then the relation g
⇐⇒

θ(g, h) > 0

(3.6)

defines a left-invariant strong order, i.e., a nonreflexive and transitive (hence, antisymmetric) binary relation on G. Therefore, the set S = {g ∈ G : θ(e, g) > 0} (3.7) is a (nonempty) semigroup, S ∩S −1 = ∅, and g < h if and only if g −1 h ∈ S. In this case, a (continuous) path L : [a, b] → (G, τ ) is said to be timelike if θ(L(t), L(s)) > 0 for all t < s in [a, b]. The length of a path is defined by the formula λ(L) = inf

n

θ(L(tk ), L(tk+1 )),

k=0

where inf is taken over all ordered partitions a = t0 < t1 < · · · < tn = b, n ∈ N, of the segment [a, b]. A left-invariant antimetric θ on (G, τ ) is said to be inner if θ(g, h) = sup λ(L) for all g, h ∈ G, where sup is taken over all timelike paths L with the endpoints g and h; moreover, if λ(L) = θ(g, h) > 0, then L is called the longest arc. Here, as usually, sup (respectively, inf) of the empty set is assumed to be equal to −∞ (respectively, +∞). Therefore, θ(G × G) ⊂ {−∞} ∪ (0, +∞) and θ(g, h) = −∞ / S (see (3.7)), if θ is a left-invariant inner antimetric on G. It is not difficult to if and only if g −1 h ∈ prove that under this condition the semigroup S is locally generated. By definition, a left-invariant inner antimetric θ on (G, τ ) is compatible with the topology τ if the function θ + (g, h) = max{0, θ(g, h)} is continuous on (G × G, τ × τ ) and the corresponding semigroup S is sharp. One can prove that for such antimetric θ, the semigroup S is open in (G, τ ), the group (G, τ ) is locally arcwise connected, and θ(G × G) = {−∞} ∪ (0, +∞) (see [20]). Theorem 9 (see [20]). Let (G, τ ) be a complete, metrizable, Hausdorff topological group. The following statements about nonnegative real-valued continuous function η on G are equivalent: (a) the ordinate set of the function η P = {(g, t) ∈ G × R : 0 < t < η(g)} is an open, sharp, locally generated subsemigroup of the group G × R; 419

(b) the formula θ(g, h) = σ(η(g −1 h)), where

t, σ(t) = −∞,

t > 0, t ≤ 0,

defines a left-invariant inner antimetric on (G, τ ) compatible with τ . There is an analog of Theorem 6 for left-invariant inner antimetrics on connected, locally compact, metrizable, Hausdorff groups (G, τ ) compatible with the topology τ , namely [20, Theorem 3.1]. Therefore, the study of such antimetrics is reduced in some sense to the study of them on connected Lie groups. Example 1 (Minkowski space-time). Let G = (Rn+1 = R×Rn , +), where n ≥ 1, is an additive vector group with the usual (Euclidean) topology τ . Let θ be a left-invariant function on G and ⎧ 1/2 n n ⎪ ⎪ ⎨ 2 2   s − xk if s > 0, s2 − x2k > 0, θ (0, . . . , 0), (s, x) = k=1 k=1 ⎪ ⎪ ⎩ −∞ otherwise. Then θ is a left-invariant inner (Lorentzian) antimetric on G compatible with τ . Moreover, if θ(g, h) > 0, g, h ∈ Rn+1 , then any homeomorphic parametrization of the rectilinear segment in Rn+1 with the endpoints g and h is the longest arc; there is no other longest arcs. In a recent paper by V. M. Gichev (see [54]) is given an elegant characterization of finite Coxeter groups and polar representations of connected, compact Lie groups in terms of subset semigroups in finite-dimensional additive vector groups. Information about polar representation presented below follows [54]. An orthogonal representation r of a compact Lie group G in a finite-dimensional Euclidean vector space V is said to be polar if there exists a linear subspace A ⊂ V (Cartan subspace) such that (a) any orbit Ov = r(G)(v), v ∈ V , intersects A; (b) at any point u in Ov ∩ A, the tangent space Tu Ov to the orbit Ov is orthogonal to A. Note that any point from Ov ∩ A is isolated in Ov , while the set Ov ∩ A is finite. In the definition of a polar representation, we do not assume that G is connected. However, the corresponding property depends only on the connected component of the unit Ge of the group G. In particular, all linear linear representations of finite groups are polar by definition. The class of polar representations was defined by J. Dadok (see [47]). A model example provides the adjoint representation of Lie group G in its Lie algebra g, where the role of A in this case is played by the Cartan subalgebra. A more general example is an s-representations, i.e., an isotropy representation of a symmetric space M = H/K used below in Theorem 18 of Sec. 10. These representations are realized in spaces V of the Cartan decomposition h = k + V , where one can take as A a maximal Abelian subspace in V . It is proved in [47] that polar representations are orbitally equivalent to s-representations; this means the existence of an embedding G → K such that G is transitive on any orbit of the group K in V . This result was deduced in [47] from the classification of polar representations. A conceptual proof for representations with orbits in general position for codimensions greater than 2 and the deduction of full classification is given in papers of J.-H. Eschenburg and E. Heintze [50, 51]. The Weyl group of a polar representation r is defined as the restriction to the Cartan subspace A of actions by those transformations from r(G) that preserve A. Below, a Coxeter group is a finite linear group generated by reflections in hyperplanes. Theorem 10 (see [54]). The set of convex hulls of orbits of an exact orthogonal representation r of nontrivial compact Lie group G is a semigroup with respect to the Minkowski addition (the sum of sets 420

B ⊂ V and C ⊂ V is the set B + C = {b + c, b ∈ B, c ∈ C}) if and only if r is polar and its Weyl group is a Coxeter group. The first condition for connected Lie group G is equivalent to the polarity of r, and if r(G) is finite, it is equivalent to the condition that r(G) is a Coxeter group. 4.

Homogeneous Geodesic Manifolds

Recall necessary information about smooth (i.e., C ∞ -) manifolds. Let M bea smooth (n-dimensional) manifold, Tx M be the tangent vector space to M at the point Tx M be its tangent vector bundle with the natural structure of a smooth manifold (of x, T M = x∈M

dimension 2n), and p : T M → M be the canonical (smooth) projection, which assigns to each tangent vector over M its origin (application point). A vector field on M is a smooth mapping X : M → T M such that p ◦ X = idM is the identical mapping of the manifold M . The set of all smooth real-valued functions f : M → R with pointwise addition and product operations forms the ring F (M ), which includes the field R of constant functions. Moreover, the set V (M ) of all smooth vector fields on M with respect to pointwise addition of their vector values and product (on the left) by functions from F (M ) is a module over the ring F (M ). The formula Xf (x) = X(x)f for X ∈ V (M ), f ∈ F (M ), x ∈ M , where X(x)f is the derivative of the function f in the direction of the vector X(x), defines the function Xf ∈ F (M ). The mapping f ∈ F (M ) → Xf ∈ F (M ) is a derivation of the ring F (M ), i.e., it satisfies the following conditions: (1)

X(f + g) = Xf + Xg,

(2)

X(f g) = (Xf )g + f (Xg),

f, g ∈ F (M ).

Conversely, any derivation of the ring F (M ) is realized by a unique element X ∈ V (M ). This allows one to define the Lie bracket [X, Y ] ∈ V (M ) of elements X, Y ∈ V (M ) as the derivation of the ring F (M ) by the formula [X, Y ] = X ◦ Y − Y ◦ X. With respect to this operation, the vector space V (M ) over the field R becomes an infinite-dimensional Lie algebra, i.e., (1) [X, Y ] = −[Y, X], (2)

[aX + bY, Z] = a[X, Z] + b[Y, Z],

(3)

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0

for any elements a, b ∈ R and X, Y, Z ∈ V (M ). Moreover, taking into account the module structure of V (M ) over F (M ), we have (4)

[f X, gY ] = f g[X, Y ] + f (Xg)Y − g(Y f )X,

where X, Y ∈ V (M ) and f, g ∈ F (M ). Definition 9. A subset D ⊂ T M is called a smooth vector subbundle (of rank k ≤ n = dim M ) of the tangent vector bundle T M of a smooth manifold M , or a distribution, or a polarization on M if (1) Dx := D ∩ Tx M is a vector subspace in Tx (M ) of dimension k for any point x ∈ M ; (2) for any point x ∈ M , there exist its open neighborhood U and vector fields X1 , . . . , Xk ∈ V (U ) such that for any point y ∈ U , the vectors X1 (y), . . . , Xk (y) are linearly independent in Dy . Any such collection X1 , . . . , Xk ∈ V (U ) is called a local basis of distribution D in U . A distribution D on M is said to be totally nonholonomic if for any point x ∈ M , there is a local basis X1 , . . . , Xk ∈ V (U ) of the distribution D in a neighborhood U of the point x such that some collection Y1 , . . . , Yn from the Lie subalgebra in V (U ) generated by the elements X1 , . . . , Xk is a local basis of the tangent bundle T U . 421

A distribution D on M is said to be integrable if for any smooth vector fields X and Y on M tangent to the distribution D, the vector field [X, Y ] is also tangent to D. Remark 3. The term “polarization” was used in [68]; in the sequel, we do not use it. More exactly, in the definition of an integrable distribution D, one must say that the images of the corresponding vector fields are contained in D; instead of this, for brevity, we say a vector is tangent (to distribution D), etc. In consequence of the Frobenius theorem (see [135]), the distribution D is integrable if and only if it is tangent to fibers of some foliation on the manifold M (see [123]). A trivial example of a totally nonholonomic distribution on a smooth manifold M is its tangent bundle T M , but if dim M ≥ 3, then M admits other totally nonholonomic distributions. One can easily prove the following proposition. Proposition 3. Let D be a totally nonholonomic distribution on a smooth manifold M and X1 , . . . , Xk , where k ≤ n = dim M , is a local basis of the distribution D in some neighborhood U of a point x ∈ M . Then there are vector fields Y1 , . . . , Yn from the Lie subalgebra (of the Lie algebra V (U )) generated by the elements X1 , . . . , Xk and a neighborhood O ⊂ U of the point x such that the restrictions of the vector fields Y1 , . . . , Yn to O form a (local) basis of the tangent bundle T O (in O). Remark 4. This proposition means that in the definition of a totally nonholonomic distribution, one can take any its local bases with possible restrictions to smaller neighborhoods. If G is a connected Lie group of dimension n and H is its compact Lie subgroup of dimension l, then the formula (4.1) σ : G × G/H → G/H, σ(g, g1 H) = gg1 H, defines a smooth left action of the Lie group G on the homogeneous manifold G/H (consisting of left cosets). Theorem 11 (see [12–14]). Every homogeneous geodesic manifold (M, ρ) is isometric to the quotient space G/H of some connected Lie group G with isotropy subgroup H supplied with a G-invariant geodesic metric. Any such metric on G/H is a (perhaps, nonholonomic) Finsler metric dF defined by a G-invariant, totally nonholonomic distribution D on G/H and a Finsler norm F on D. The distance dF (x, y) between points x, y ∈ G/H is defined as the greatest lower bound of the lengths (integrals of lengths of tangent vectors, measured by the norm F ) of piecewise smooth paths in G/H joining points x and y and tangent to the distribution D. Conversely, any such pair (D, F ) on G/H defines a metric dF on G/H by the way indicated above, and the metric space (G/H, dF ) is a homogeneous geodesic manifold. Remark 5. We will say that the metric dF defined by a totally nonholonomic distribution D on a smooth manifold M and a (continuous) Finsler norm F on D is a nonholonomic Finsler metric only in the case where D = T M . If, in addition, the norm F is Euclidean, then the metric dF is called a Carnot–Caratheodory metric or a sub-Riemannian metric. Later, we will say that the metric ρ = dF is a Finsler metric if D = T M . Note that we assume, as well as H. Busemann in the book [39], that a Finsler manifold is a finite-dimensional C 1 -differentiable manifold M with continuous norm F on its tangent bundle T M , while specialists in Finsler geometry usually burden this notion with additional requirements for the norm F (see [52, 114]). Remark 6. In [21, Example 16], V. M. Gichev presented a left-invariant “nonholonomic” inner antimetric θ (defined in Sec. 6) on the Heisenberg group H 3 compatible with its topology, which may be called sub-Lorentzian, and indicated all its longest arcs. In this case, any two elements g, h ∈ H 3 such that θ(g, h) > 0 can be joined by the longest arc. Later results about left-invariant sub-Lorentzian structures on Lie groups are obtained in [66, 85]. 422

We make a remark on Theorem 11 and its proof. The group I(M ) of all motions of a homogeneous manifold (M, ρ) is a Lie group (see, e.g., [122]). As the group G in Theorem 11, one can take the connected component of the unit of the Lie group I(M ) or any closed connected subgroup in I(M ), which is transitive on M , and as the group H, the stabilizer of any point x0 ∈ M in the group G. Then the mapping f : G/H → M,

f (gH) = gH(x0 ) = g(x0 ) = x,

g ∈ G,

x ∈ M,

is a homeomorphism of the homogeneous manifold G/H of the Lie group G on the manifold (M, ρ). Under the identification of (M, ρ) with G/H by means of homeomorphism f −1 , the action of Lie group G on M turns into the left action of the group G on G/H, defined by formula (4.1). It is easier to get the proof of the last statement in Theorem 11. It is based on the fact that in papers of P. K. Rashevsky [113] and W. L. Chow [45] they proved that any two points in connected smooth manifold M can be joined by some piecewise smooth path, which is tangent to a totally nonholonomic distribution D on M (so-called horizontal path). The proof of other statements in Theorem 11 is conducted in two steps. Later, we will need, in addition, some well-known notions and notation. Let p : G → G/H be the canonical projection defined by formula p(g) = gH,

g ∈ G,

and lg , rg : G → G be the left and right shifts of the group G by an element g ∈ G defined by formulas lg (g ) = gg ,

rg (g ) = g g,

and dp : T G → T (G/H), be their differentials. The formula

d(lg ), d(rg ) : T G → T G

C2 = {v ∈ T G : dp(v) = 0}

(4.2)

defines an integrable distribution on G, which is invariant under the shifts lg , g ∈ G, and rh , h ∈ H, i.e., d(lg )(C2 ) = C2 , d(rh )(C2 ) = C2 . The set LG of all left-invariant vector fields X on G such that X ◦ lg = d(lg ) ◦ X for all g ∈ G forms a Lie subalgebra of dimension n of the Lie algebra (V (G), [·, ·]), while the set LH of all elements in LG that are tangent to the left-invariant distribution C2 is a Lie subalgebra in LG of dimension l. The Lie algebra LG (respectively, LH) is called the Lie algebra of the Lie group G (respectively, of the Lie group H). It is clear that if X ∈ LG and Y ∈ LH, then X(g) = d(lg )(X(e)),

Y (g) = d(lg )(Y (e)),

where e is the unit of the group G, X(e) ∈ Te G, and Y (e) ∈ Te H. These formulas define isomorphisms of the vector spaces LG (respectively, LH) and Te G (respectively, Te H). Transferring the structure of Lie algebras by these isomorphisms from LG and LH to Te G and Te H, it is convenient to identify the Lie algebras LG and LH with Te G and Te H, respectively. First, in [13] is considered a homogeneous manifold (M, ρ) with a simply transitive Lie group G of motions, i.e., a connected Lie group G with left-invariant geodesic metric ρ. Let us recall that a (compact) closed ball Br = B(e, r), r ≥ 0, possesses properties B1–B5. Let L = LG be the Lie algebra of the Lie group G. The exponential mapping exp : L → G defines a 423

diffeomorphism of some neighborhood V of zero in L onto a neighborhood U = U (e, r0 ), r0 > 0, of the element e ∈ G. For a fixed number r, where 0 < r < r0 , and a number k ∈ N ∪ {0}, let us define the set βk = 2k exp−1 (Br/2k ). One can subsequently prove the following statements: (1) the sequence of compact subsets βk , k ∈ N ∪ {0}, is not increasing, i.e., βk+1 ⊂ βk for all natural numbers k;  βk is a compact, centrally symmetric, convex subset in V centered at 0 ∈ V ; (2) the set β := k≥0  (3) the linear hull c := λβ of the set β is a vector subspace of the Lie algebra L such that the λ>0

least Lie subalgebra ofLie algebra L, including c, coincides with L; (C(g) = d(lg )(c)) on G is left-invariant and totally nonholonomic; (4) the distribution C = g∈G

(5) if F is an left-invariant norm on C such that β = {rv|v ∈ c, F (v) ≤ 1}, then the (perhaps, nonholonomic) Finsler metric dF on G corresponding to the pair (C, F ) coincides with ρ. The proof for the case of a G-invariant geodesic metric ρ on G/H is contained in [14]. First, it was proved that there exists a geodesic metrics ρ1 on G, which is invariant under all shifts lg , g ∈ G, and rh , h ∈ H, and is such that the canonical projection p : (G, ρ1 ) → (G/H, ρ) is a submetry. In a consequence of results stated above, the metric ρ1 coincides with some (perhaps, nonholonomic) Finsler metric dF1 defined by a pair (C, F1 ) on G, invariant under all shifts lg , g ∈ G, and rh , h ∈ H, where C is a totally nonholonomic distribution on the connected Lie group G and F1 is a norm on C. The metric ρ1 can be chosen in such a way that the distribution C includes the distribution C2 on G mentioned above. After this, it was proved that a (well-defined) G-invariant distribution D = dp(C) on G/H is totally nonholonomic (see also Theorem 16 below). The formula F (v) = min{F1 (u)|u ∈ C, dp(u) = v},

v ∈ D,

defines a G-invariant norm on the distribution D ⊂ T (G/H). Finally, it was proved that ρ = dF . E. Le Donne [91] proved the following bi-Lipschitz version of Theorem 11. Theorem 12. Let G be a Lie group and H be a compact subgroup of G. Further, let M = G/H be corresponding homogeneous space supplied with a geodesic metric d that induces its natural topology. Assume that there is a subgroup GS of the group G acting transitively on M by locally bi-Lipschitz mappings relative to d. Then there exists a totally nonholonomic GS -invariant distribution such that any Carnot–Caratheodory metric corresponding to this distribution is locally bi-Lipschitz equivalent to the metric d. Let us show how to prove statement 3 of the Introduction. Consider a nonholonomic Finsler (respectively, Riemannian) space (G/H, dF ) defined by a Ginvariant totally nonholonomic distribution D = T (G/H) on G/H and a Finsler norm F (defined by a scalar product) on D. There are a G-invariant distribution D1 on G/H such that T (G/H) = D ⊕ D1 and a G-invariant norm F1 (defined by the scalar product) on D1 . Define the family of G-invariant norms Ns , s > 0, on T (G/H) by the formulas Ns (w) = F (v) + sF1 (v1 ) if

w = v + v1 ,

where v ∈ D, v1 ∈ D1 .

Every norm Ns defines a homogeneous Finsler (respectively, Riemannian) manifold (G/H, ρs ). One can prove that there exists the Gromov–Hausdorff limit of the homogeneous manifolds (G/H, ρs ) (with monotonically increasing metrics) as s → +∞, and this limit is isometric to the space (G/H, dF ). (Note that V. Ya. Gershkovich called the method applied here the method of penalty metrics.) 424

Remark 7. An analogous statement is valid for any nonholonomic Finsler or sub-Riemannian manifold. On the grounds of this analog and the famous work of John Nash [106], E. Le Donne proved in [92] that every connected sub-Riemannian manifold M admits a topological embedding f into some finite-dimensional Euclidean space E k such that the mapping f : M → f (M ) is an isometry if the distance between any two points in f (M ) ⊂ E k is defined as the greatest lower bound of the lengths of paths in f (M ) that join these points. Now let us show how the same manifold (G/H, dF ) can be obtained as the Gromov–Hausdorff limit of the spaces (G, ρs ) with left-invariant nonholonomic Finsler (respectively, Riemannian) metrics ρs on G as s → 0. Let p : G → G/H be the canonical projection and C = dp−1 (D). The distribution C is totally nonholonomic (see Theorem 16). Since the subgroup H is compact, there is some G-leftinvariant and H-right-invariant decomposition C = C1 ⊕ C2 into the direct sum of distributions, where C2 is defined by formula (4.2). The formula F1 (v1 ) = F (dp(v1 )) correctly determines a G-left-invariant and H-right-invariant norm on C1 defined by the scalar product if F is a Euclidean norm on D. There is a G-left-invariant and H-right-invariant norm F2 on C2 defined by a scalar product. We define the family of G-left-invariant and H-right-invariant norms ns , s > 0, on C by the formulas ns (v) = F1 (v1 ) + sF2 (v2 ) if

v = v1 + v2 ,

where v1 ∈ C1 , v2 ∈ C2 .

Every norm ns defines a homogeneous nonholonomic Finsler (respectively, Riemannian) manifold (G, ρs ), and, moreover, the canonical projection p : (G, ρs ) → (G/H, dF ) is a submetry. Now it is not difficult to prove the statement formulated at the beginning of this paragraph. The method applied here can be called the method of stimulating metrics. Now statement 3 of the Introduction is a consequence of two statements that have been proved here and Theorems 3, 4, and 11. Definition 10. A bijection of a metric space (M, ρ) onto itself is called a (metric) a-similarity, where a ∈ R, a > 0, if ρ(f (x), f (y)) = aρ(x, y) ∀pointsx, y ∈ M. An a-similarity is said to be nontrivial if a = 1. Theorem 13. A homogeneous locally compact space with inner metric (M, ρ) admits nontrivial metric similarities if and only if (M, ρ) is isometric to a finite-dimensional normed vector space or a Carnot group, i.e., a connected, simply connected, graduated, nilpotent (noncommutative) Lie group G with m  Lk (of nilpotence length m > 1), which is the direct sum of vector the Lie algebra LG = L = k=1

subspaces Lk ⊂ L with conditions [Li , Lj ] = Li+j ,

Lk = 0 if k > m;

with a left-invariant nonholonomic inner metric dF defined by some left-invariant norm F on the left-invariant distribution D, where D(e) = L1 . Remark 8. Apparently, the term “Carnot group” first appeared in [110]. In the sequel, we discuss some properties of Carnot groups. The first author knew of the results of H. Radstr¨ om’s paper [112] much later than his publications [13, 14]. Meanwhile, [112, Theorem 4.12] and the method of its proof are similar in many respects to results and methods from [13] and constitute a good addition to them. Let us give a direct quotation of this theorem and the definition from [112] necessary for its formulation. 425

Definition 11. We say that a compact subset K of the Lie algebra LG of a Lie group G generates a one-parameter semigroup φ of subsets in G if for any number γ ≥ 0, the equality φ(γ) = lim (exp(δK))[γ/δ] δ→0

(4.3)

holds, where [γ/δ] is the integer part of the number γ/δ. Note that it is necessary to introduce an arbitrary auxiliary left-invariant (possibly, Riemannian) metric ρ on G whose topology coincides with the usual topology of the Lie group G. Then the family of all compact subsets in G with the corresponding Hausdorff metric H(ρ) constitutes a complete metric space (K(G), H(ρ)) (see [89]). The limit on the right-hand side of formula (4.3) is understood as the limit of elements in (K(G), H(ρ)). Theorem 14. Let G be a Lie group and LG be its Lie algebra. Then (1) every compact subset in LG generates a one-parameter semigroup φ of subsets in G such that φ(0) = {e}; (2) every compact subset in LG generates the same semigroup as its convex hull; (3) every one-parameter semigroup φ of subsets in G with condition φ(0) = {e} is generated by a unique compact and convex subset in LG; (4) a compact and convex set K generating a given semigroup φ is defined by the equality 1 exp−1 (φ(δ)). δ→0 δ

K = lim

(4.4)

Note that in the hint of the proof of the main result for Lie groups from [13] given above, the 1 semigroup Bs , s ≥ 0, is generated by the set β. r As a corollary of Theorem 14, H. Radstr¨ om obtained an exact description of one-parameter semigroups of compact sets in Lie groups in [112, Theorem 4.13], but only in the simplest commutative case. Theorem 15. Let G be a commutative Lie group and LG be its Lie algebra. Further, let a compact and convex set K ⊂ LG generate a one-parameter semigroup φ of sets in G. Then φ(δ) = exp(δK). Remark 9. H. Radstr¨om’s remark given immediately after this theorem is very interesting: “We have not much general information about the structure of the sets φ(δ) defined by a one-parameter semigroup (of sets) in a Lie group. The theorem just proved shows that at least in the commutative case, these sets are images of compact convex sets relative to continuous open mappings, and that if sufficiently small sets are even homeomorphic to such sets. Although it seems plausible that this is true in general Lie groups also, I have not succeeded in proving or disproving it.” Most likely, under “compact convex sets” are meant generating compact convex sets of considered one-parameter semigroups of sets. Then H. Radstr¨om’s conjecture is valid only for commutative Lie groups and Lie groups Gn , n ≥ 2, with Lie algebras Ln considered in Sec. 9, and is invalid in the general case. Every Lie group Gn , n ≥ 2, is characterized by the property that any left-invariant Riemannian metric on it has constant negative sectional curvature. In any case, it is very likely that Radstr¨om could have proved all the main results of the paper [13] 36 years before its publication if he knew of the results of P. K. Rashevsky [113] and W. L. Chow [45] published before. In the paper [1], H. Abels and G. A. Margulis introduced an interesting notion of coarsely geodesic metric, which generalizes the notions of inner and geodesic metrics. We shall not give here the definition of this metric but restrict ourselves to an example of such metric and a problem naturally connected with this example given in [1]. 426

Let U be an open, bounded (i.e., the closure U is compact), symmetric (i.e., U −1 = U ) neighborhood of the unit in a connected Lie group G. Define the corresponding “word” metric on G by the formula ρU (g, h) = min{i ≥ 0 | g−1 h ∈ U i }.

(4.5)

Obviously, the metric ρU is left-invariant and defines the discrete topology on G. On the other hand, it is coarsely geodesic and quasi-isometric to any left-invariant Riemannian metric ρ on G (see [1]); it is not difficult to prove that one can take as ρ any left-invariant inner metric on G. Recall that the metrics ρ1 and ρ2 on the set M are called quasi-isometric if there is a number C > 1 such that 1 ρ2 (x, y) − C ≤ ρ1 (x, y) ≤ Cρ2 (x, y) + C for all points x, y ∈ M . C Metrics ρ1 and ρ2 on a set M are said to be coarsely equal if there exists a number C > 0 such that |ρ1 (x, y) − ρ2 (x, y)| ≤ C

for all points x, y ∈ M

(see [1]). Problem 3. Let U be a connected, open, bounded, symmetric neighborhood of unit in an arbitrary connected Lie group G and ρU be the corresponding left-invariant metric on G defined by formula (4.5). Is there a left-invariant inner metric ρ on G that is coarsely equal to the metric ρU ? 5.

On Nonholonomic Metric Geometry

This section with short (known to the authors) information about nonholonomic metric geometry and its development is a natural continuation of the last comments on H. Radstr¨ om’s paper. Objects of study in nonholonomic metric geometry are Riemannian or Finsler manifolds (M, (·, ·)) or (M, F ) with smooth totally nonholonomic (horizontal) distribution D on them of rank less than the dimension of manifold. Then on any such manifold, one can naturally define a (nonholonomic) inner metric ρD , which is the distance between any two points defined as the greatest lower bound of the lengths of piecewise smooth horizontal (i.e., tangent to the distribution D) paths that join given points (note that the lengths are measured in the ambient manifold). The existence of such paths, and consequently, the finiteness of such metric is a consequence of the Rashevsky–Chow theorem (see [45, 113]). It is clear that in this case, one can set a scalar product (·, ·) or a norm F only on the distribution D. But sometimes it is convenient to introduce a Riemannian or Finsler structure on the whole manifold. For example, if the space with inner metric corresponding to the space (M, (·, ·)) or (M, F ) is complete and consequently geodesic, then the metric space (M, ρD ) is also complete and geodesic. A particular case of a distribution D is a contact structure, i.e., a totally nonholonomic hyperdistribution on a manifold M of dimension 2k + 1 defined as the zero set D = {v ∈ T M | ω(v) = 0}

(5.1)

of a contact differential 1-form (Pfaffian form) ω, i.e., satisfying the condition ω ∧ (dω)k = 0, where (dω)k denotes the kth exterior degree of the form dω. For unclear reasons, the history of the development of nonholonomic metric geometry (for details on this history, see the survey [129] of A. M. Vershik and V. Ya. Gershkovich) consists of loosely connected episodes. Only during the last 30 years did it develop systematically and sufficiently rapidly, also giving essential results obtained by predecessors. One can find some separated results connected with nonholonomic geometry in papers by D. M. Sintsov of 19th and 20th centuries (see, e.g., [116]). The actual origin of the theory is the classical paper of C. Caratheodory [40] on the axiomatization of thermodynamics, connected with the sixth Hilbert problem on a mathematical presentation of physical axioms (see [76]). In this paper, for the introduction of entropy under the hypothesis of the 427

second law of thermodynamics in his formulation, Caratheodory actually proved a particular case of the Rashevsky–Chow theorem for a contact distribution D, where horizontal curves correspond to adiabatic processes, if one assumes that the identity Q ∧ dQ ≡ 0 for the heat 1-form Q is violated. A left-invariant totally nonholonomic distribution on the Heisenberg group, which is inconsistent with its tangent bundle, is connected with thermodynamics equations, in particular, with equations of the French physicist Sadi Carnot. He was the son of the outstanding politician and remarkable mathematician Lazar Carnot, a member of the Directory, military minister in Napoleon’s government, his general, and at the same time the author of mathematical works in abstract geometry, logical foundation of analysis, and of the work “The main principles of equilibrium and movement” published at 1803. Sadi Carnot published in 1824 the first paper in thermodynamics [42]. We shall speak in more detail of the papers of S. Carnot and C. Caratheodory in the next section. According to what has been said, M. Gromov called a nonholonomic Riemannian metric a Carnot–Caratheodory metric, and the corresponding metric spaces Carnot–Caratheodory spaces (see [68]). The other titles of nonholonomic metric and spaces are singular Riemannian (see [73]) or sub-Riemannian (see [117]). Note in passing that W. L. Chow was C. Caratheodory’s disciple, and even his main works are in algebraic geometry. The dissertation of the German and Swiss mathematician and astronomer P. Finsler, republished much later as the book [52], was written in 1918 and defended at 1919 in G¨ottingen under the supervision of C. Caratheodory. Already in 1928 J. L. Synge obtained (see [120]) differential equations of “nonholonomic geodesics” in Riemannian geometry, i.e., of stationary admissible (tangent to a given distribution D) curves for the length functional. The possibility of the introduction and use of left-invariant nonholonomic Riemannian and Finsler metrics on linear Lie groups in relation to linear control system, without any mention of the papers of Rahsevsky and Chow, is outlined in the paper [37] of R. W. Brockett: “In problems that are interesting in systems theory, the generalization of the usual approach to such questions as existence of geodesics, expressions for Laplacian spectrum etc. is often required. Such generalization, nonstrictly speaking, is developed for inclusion in a number of possible metrics, infinite in some direction and subject only to the condition that the directions in which it is finite can be joined so that the distance between any two points is finite.” Generally speaking, one must note the close relation of the theory of left-invariant inner metrics ρ on Lie groups G, among them nonholonomic, to contemporary development of mathematical control theory (including optimal). In [12] the following method for searching the shortest arcs on (G, ρ) is suggested. Any rectifiable path x = x(t) on (G, ρ) parameterized by the arc length t, satisfies almost everywhere the ordinary differential equation (control system) x(t) ˙ = d(lx(t) )(u(t)),

u(t) ∈ U,

(5.2)

with a measurable (generally, discontinuous) control function u(t) and a control region U , which is the unit ball of the corresponding Riemannian or Finsler norm on D(e) ⊂ LG = Te G. The length of such path x = x(t) is equal to the time of motion. Therefore, any shortest arc in (G, ρ) must be a time-optimal trajectory of the control system (5.2) and satisfy the Pontryagin maximum principle (see, e.g., [111]). This principle consists of the statement that for a time-optimal trajectory x(t) of system (5.2), the “Hamiltonian function” H(x(t), ψ(t), u(t)) defined by this trajectory must attain the maximum M (t) with respect to u(t) ∈ U for almost all t; moreover, it turns out that M = M (t) is constant. The metric ρ is finite if and only if any two points in G can be joined by a trajectory with an equation of the form (5.2). In the language of control theory, this means that system (5.2) is attainable or controllable. Note that the distance ρ(x, y) is usually called the Bellman function. All these statements (except for the special form of Eq. (5.2) related to the left invariance of the metric on Lie group) are applicable to all smooth sub-Riemannian manifolds (see [93]). 428

In the next section, we discuss, in more detail, methods of searching for the geodesics on homogeneous manifolds with inner metric. We mention R. Hermann’s paper [74] on the controllability of nonlinear systems. In 1970s, many investigations about controllable nonlinear systems appeared; we mention only the paper of H. J. Sussman [119], closely related to the Rashevsky–Chow theorem. There are several very useful books on geometric control theory (including optimal), for example, by V. Jurdjevich [82] and by A. A. Agrachev and Yu. L. Sachkov [3]. M. Gromov’s paper [67] greatly influenced the extension of the investigations to nonholonomic metric geometry. This paper is dedicated to the Milnor–Wolf problem related to the growth of volumes of balls for universal coverings of Riemannian spaces and investigations of fundamental groups of compact Riemannian manifolds with some restrictions on curvature: Is it true that a finitely generated (discrete) group has polynomial growth if and only if the group is virtually nilpotent, i.e., contains a nilpotent subgroup of finite index, and otherwise has exponential growth? In [67], the following assertion is proved: If a finitely generated subgroup Γ has polynomial growth, then Γ contains a nilpotent subgroup of finite index. Let Γ be a group with generators {γ1 , . . . , γk }. Then every element γ ∈ Γ can be represented in the form of a word γ = w = γip11 γip22 · · · γipll ,

where p1 , . . . , pl are integers; the number |p1 | + · · · + |pl | is called the length of the word w. The norm |γ| is defined as the minimal length of words that represent the element γ. It is easy to see that |γ| = |γ −1 |,

|γ1 γ2 | ≤ |γ1 | + |γ2 |.

If {δ1 , . . . , δm } is another generator system in Γ, then the norm | · |1 corresponding to it does not necessarily coincide with | · |. But there is a number C ≥ 1 such that 1 |γ|1 ≤ |γ| ≤ C|γ|1 (5.3) C for all γ ∈ Γ. The formula r(γ1 , γ2 ) = |γ1−1 γ2 | defines a left-invariant metric d on Γ. We denote by B(m), m ≥ 0, the ball of radius m centered at e ∈ Γ relative to the metric d, and by |B(m)| the (finite) number of elements in the ball B(m). The group Γ is said to have polynomial growth if there are positive numbers d and c such that |B(m)| ≤ cmd for all balls B(m), m ∈ N; the group Γ has exponential growth if |B(m)| ≥ K m for some number K > 1. It follows from formula (5.3) that these definitions are independent of the choice of generators, and consequently, the corresponding notions are well defined for all finitely generated groups. If Γ is nilpotent, then Γ has polynomial growth (see [139] or the Appendix to [67] by J. Tits). At first, M. Gromov noted in [67] that if Γ is a finite extension of a group of polynomial growth, then Γ itself has polynomial growth. He concludes from here: If a finitely generated group Γ has nilpotent subgroup of finite index, then Γ has polynomial growth. After that, M. Gromov began the proof of the converse statement and obtained at the end a positive answer to the first part of the Milnor–Wolf problem. We are especially interested in the method of the proof. M. Gromov applies in the proof the Milnor–Wolf theorem (see [99, 139]): A finitely generated 429

solvable group Γ has exponential growth if Γ contains no nilpotent subgroup of finite index (here Gromov erroneously refers to another paper of J. Milnor from the same journal). J. Tits notes in the Appendix to [67] that in the proof, it suffices to use the following partial case of the Milnor–Wolf theorem: Let L be a group. Assume that there is a homomorphism ξ of the group L onto Z whose kernel is finitely generated and virtually nilpotent. Then L itself is virtually nilpotent or has exponential growth. He proves this statement in [67, Lemma 2]. After several algebraic lemmas, M. Gromov proved that there exists a sequence of positive numbers εn → 0 such that (1) there exists the Gromov–Hausdorff limit (see [67]) of the sequence of metric spaces εn Γ := (Γ, εn r); (2) this limit is isometric to some finite-dimensional normed vector space or some Carnot group with left-invariant nonholonomic inner metric (G, dF ) described in Theorem 13 of this review. Note that in this case, the metric dF is not sub-Riemannian, i.e., the norm is not Euclidean. Moreover, it is clear that the metric limit from statement (1) admits nontrivial metric similarities. Therefore, statement (2) follows from statement (1) and Theorem 13. There remains one unsolved question from the Milnor–Wolf problem posed by Milnor in more exact form as Problem 5603 in [41]: Is it true that any finitely generated group may have only polynomial or exponential growth? R. Grigorchuk gave a negative answer to this question, constructing finitely generated groups of intermediate growth, i.e., groups whose growth functions are neither polynomial nor exponential. The announcement of the result and its proof have been published respectively in [64] and [65]. The groups constructed by Grigorchuk have also many other unusual properties. Theorem 13 is also important with respect to the fact that under the regularity condition of the distribution D (see Definition 12 below), at any point x ∈ (M, ρ) of a nonholonomic Finsler or Riemannian manifold, there exists a tangent cone τx M to the manifold (see [67]) defined as the Gromov–Hausdorff limit of spaces ((M, x), αρ) as α → +∞, isometric to some (metrized) Carnot group Gx . This notion of the tangent cone was used by G. Margulis and G. Mostow in [96]. P. Deligne noted that under such definition, the tangent cone is not defined up to a unique isomorphism. In response to this criticism, in [97] an inner definition of tangent cone τx M is presented for a (regular) sub-Riemannian manifold M as a collection of some equivalence classes of smooth paths in M with the origin x rectifiable at x (i.e., rectifiable on some initial segment). Margulis and Mostow specified that their paper is based on ideas of L. Rothchild–E. Stein, R. Goodman, G. Metivier, M. Gromov, and J. Mitchell, and that they mainly reorganize known results rather than introduce new methods. It was proved in the doctoral dissertation of the first author that any homogeneous manifold with inner metric at any its point has a tangent cone independent of the point, which is isometric to a normed vector space or a Carnot group with left-invariant nonholonomic Finsler metric (not published). This fact underlies the proof of Theorem 13. Definition 12. A smooth distribution D = D1 on a smooth manifold M is said to be regular if there is a sequence of smooth vector subbundles with conditions D 1 ⊂ D 2 ⊂ · · · ⊂ D k = T M,

D i+1 = Di + [D i , D 1 ],

i ≥ 1,

(5.4)

where all inclusions are strict. In this case, the vector (l1 , . . . , lk ) = (dim(D 1 ), . . . , dim(D k )) is called the growth vector of distribution D and the number k is called the nonholonomy degree of the distribution D. Obviously, the regularity condition for a distribution guarantees its total nonholonomy, and every G-invariant distribution on homogeneous manifold G/H is regular. Let D be a regular distribution on M and g1 = D1 , 430

gi+1 = D i+1 /D i ,

i ≥ 1,

L = g1 + · · · + gk .

(5.5)

Then the commutation operation for vector fields that are tangent to distributions D 1 , . . . , D k , induces on L the structure of infinite-dimensional nilpotent Lie algebra of degree k; moreover, the structure of the Lie algebra is defined pointwise. The Lie algebra Lx , x ∈ M , is exactly the lk -dimensional Lie algebra of the above-mentioned Carnot group Gx . The described procedure of obtaining a Carnot algebra Lx , x ∈ M , is called nilpotentization (of the distribution D). Let us give some definitions and results from paper [2] of A. A. Agrachev and A. Marigo. The Lie algebra L of the Carnot group G is called the Carnot algebra; its bidimension is the pair (dim L, dim L1 ) (see Theorem 13). A Carnot algebra is called rigid if it is isomorphic to any of its small deformations [131] in the space of Carnot algebras with given bidimension. In paper [2] is given a complete classification of rigid Carnot algebras. Besides free nilpotent algebras, there are two infinite series and 29 exceptional rigid algebras of sixteen exceptional bidimension. As the authors write, the main motivation for them to study Carnot algebras was their role as local nilpotent approximations to regular vector distributions. A smooth real manifold M with a smooth vector subbundle D ⊂ T M and a smooth complex structure J on D is called an (abstract) Cauchy–Riemann or CR-manifold (see [124]). The common complex dimension of the vector spaces D(x), x ∈ M , is called the CR-dimension of the manifold M and is denoted by CR dim(M ). A CR-manifold M is said to be regular if the distribution D is regular. Moreover, the Lie algebra Lx is usually called the Lie–Tanaka algebra of the manifold M at the point x. In addition to the usual properties common for all Carnot algebras, the Lie–Tanaka algebra L := Lx satisfies the condition [JX, JY ] = [X, Y ],

X, Y ∈ g1 (x).

(5.6)

A regular CR-manifold is said to be strongly regular if all Lie–Tanaka algebras Lx , x ∈ M , are isomorphic. Recall that the generalized Heisenberg group H 2n+1 , n ≥ 1, is defined as a connected simply connected Lie group such that there is a basis {X1 , Y1 , . . . , Xn , Yn , Z} of its Lie algebra h2n+1 , whose nontrivial brackets have the form [Xk , Yk ] = −[Yk , Xk ] = Z,

k = 1, . . . , n.

Every generalized Heisenberg group H 2n+1 , n ≥ 1, admits a natural left-invariant strongly regular CR-structure with hyperdistribution D and a positive definite Hermite quadratic Levi form F on g 1 = Cn with values in C = Cg 2 such that [u, v] = Im F (u, v),

u, v ∈ g1 ,

which exists due to the condition (5.6). Thus, a left-invariant sub-Riemannian metric dF is defined on H 2n+1 (which is a Carnot group). It is proved that any Carnot algebra with condition (5.6) is a Lie–Tanaka algebra of some strongly regular CR-manifold. Note that a regular nonholonomic Finsler (in particular, Riemannian) space (M, dF ) defined by the pair (D, F ) is fractal, i.e., its Hausdorff dimension dimH (M ) is greater than its Lebesgue dimension dimL (M ), namely, k j(lj − lj−1 ) > lk = dimL (M ), dimH (M ) = j=1

where (l1 , . . . , lk ) is a growth vector of distribution D and l0 = 0 (see [68]). For homogeneous manifolds with geodesic metric, there arise natural questions on the separation by simple geometric conditions of the following manifolds: (a) sub-Riemannian manifolds among nonholonomic Finsler manifolds; (b) Finsler manifolds; (c) Riemannian manifolds among Finsler manifolds. 431

The last two questions are not difficult to solve. Namely, a homogeneous manifold with a geodesic metric is a Finsler manifold if and only if its Hausdorff dimension coincides with its topological (i.e., Lebesgue) dimension. A homogeneous Finsler manifold is a Riemannian manifold if and only if between any of its shortest arcs with fixed joint origin there exists an angle in the sense of A. D. Aleksandrov. The authors do not know simple geometric conditions to solve the problem (a), besides characterization of homogeneous sub-Riemannian manifolds as fractal homogeneous manifolds, which are the Gromov–Hausdorff limits of homogeneous Riemannian manifolds. It is interesting that there is a probabilistic approach to the solution of this problem, at least the left-invariant inner metric on Lie groups. Omitting details, we say that it is possible to set a one-to-one correspondence between left-invariant (sub-) Riemannian metrics on a given connected Lie group G and (hypo)elliptic symmetric generated operators ((sub-)Laplacians) L of one-parameter (symmetric) Gaussian convolution semigroups of absolutely continuous (relative to the Lebesgue–Haar measure) probability measures {μt , t ≥ 0} on G (exact definitions are given ny H. Heyer in [75]). Note that Heyer does not mention at all subRiemannian manifolds. In [8], A. Bendikov and L. Saloff-Coste, with the help of Theorem 1 from this review, proved that a locally compact, connected topological group G admits a one-parameter Gaussian convolution semigroup of absolutely continuous (with respect to the Haar measure) probability measures {μt , t ≥ 0} if and only if G is locally connected and metrizable. Structures and notions of nonholonomic metric geometry naturally arise not only in thermodynamics and branches of mathematics mentioned above, but also in algebra, analysis (in particular, in the theory of quasiconformal mappings of sub-Riemannian manifolds), the theory of hypoelliptic differential equations, nonholonomic mechanics, mathematical physics, neurobiology, etc. (for example, see R. Montgomery’s book [102]).

6.

The Search for Geodesics in Homogeneous Manifolds with Inner Metric

As was noted above, for any homogeneous manifold with inner metric (G/H, dF ), there exists a left-invariant inner metric dF1 on the Lie group G such that the canonical projection p : (G, dF1 ) → (G/H, dF ) is a submetry. Therefore, any shortest arc of the space (G/H, dF ) that joins points g1 H, g2 H ∈ G/H is the image under the projection p of some shortest in (G, dF1 ) junction of the point g1 with the fiber-preimage p−1 (g2 H) = g2 H ⊂ G, and conversely. Therefore, the method of search for (locally) shortest arcs on (G, F1 ) from the paper [12] (mentioned in the preceding section) allows one to find geodesics, i.e., locally shortest arcs, also on the manifold (G/H, dF ). In [62], I. A. Gribanova using the Pontryagin maximum principle found all geodesics of an arbitrary left-invariant Finsler metric on the group G2 . In [16], by this method, geodesics are found for any leftinvariant nonholonomic Finsler metric ρ on the simplest Carnot group, namely, the three-dimensional Heisenberg group H 3 , i.e., the Lie group of real upper triangular (3 × 3)-matrices with units on the main diagonal. The Pontryagin maximum principle yields a necessary condition, and hence it misses no shortest arcs, but can give, in addition to them, some paths that are not shortest. But usually geometric considerations allow one to remove “unnecessary” paths and to find all geodesics, as was done, for example, in [16, 62]. Other methods of the search for geodesics on sub-Riemannian manifolds were suggested by A. M. Vershik and V. Ya. Gershkovich [129] and R. Strichartz [117]. In [129] and other papers of these authors, they consider the so-called “mixed vector bundle” D ⊕ D ⊥ ⊂ T M ⊕ T ∗ M over M or “centaur,” where the vector space D ⊥ (x) for x ∈ M is the annihilator of the space D(x) ⊂ Tx M , i.e., the set of all covectors ξ ∈ Tx∗ (M ) such that ξ(v) = 0 for all vectors v ∈ D(x). For any regular totally nonholonomic distribution D on a Riemannian manifold (M, (·, ·)), the Euler–Lagrange equation defines some vector field X on the mixed bundle D ⊕ D ⊥ . If 432

(M, (·, ·)) is complete, then a vector field X defines a globally defined flow Φ : D ⊕ D⊥ × R → D ⊕ D⊥ consisting of integral curves of the vector field X. This flow is called nonholonomic geodesic flow and the canonical projections of integral curves of the vector field X to M are called nonholonomic geodesics. R. Strichartz in [117] defined a sub-Riemannian metric on the pair (M, D) as a positive-definite quadratic form Q(x) on D(x) smoothly depending on x ∈ M . To the form Q(x) corresponds a linear mapping g(x) : Tx∗ M → Tx M defined by the condition   Q(x) v, g(x)(ξ) = ξ(v) for ξ ∈ Tx∗ M and all v ∈ D(x). In addition, ker g(x) = D ⊥ (x). The converse statement is also valid. Therefore, we can suppose that the family of mappings g(x), x ∈ M , defines a sub-Riemannian metric on (M, D) in the sense presented above. After this, we define the Hamilton function  1  H(x, ξ) = ξ g(x)(ξ) , ξ ∈ Tx∗ M, 2 and the corresponding Hamilton–Jacobi equations (HJ) for paths in T ∗ M , which can be written in the symbolic form (HJ) x˙ = ∇ξ H, ξ˙ = −∇x H. A C 2 -path in M is called a geodesic if there exists its cotangent lift satisfying the Hamilton–Jacobi equations (HJ). Apparently, the Vershik–Gershkovich method and the Strichartz method are equivalent, but the proof is unknown to the authors. Both methods are based on Lagrange multipliers. It follows from [93, Appendix C] or the book of E. B. Lee and L. Marcus [90] that nonholonomic geodesics from [129] and geodesics from [117] on smooth sub-Riemannian manifolds are locally shortest arcs. At the same time, in contrast to ordinary Riemannian manifolds, in the general case the converse statement is not valid for sub-Riemannian manifolds. This can be confirmed by (counter)examples of sub-Riemannian manifolds that admit so-called (strictly) abnormal geodesics (i.e., geodesics for which the Lagrange multiplier λ0 , corresponding to the length functional vanishes); for more details see [93]. The method of Lagrange multipliers is not applicable to them. In particular, these examples disprove [117, Corollary 6.2] and the main theorems from [7, 69], which state that any curve in a sub-Riemannian manifold (locally) minimizing the length is a geodesic in the sense of [117]. The first counterexample was constructed by R. Montgomery (see [100]). In this counterexample, a sub-Riemannian metric on R3 is considered; the corresponding totally nonholonomic distribution is irregular and has rank two (a simpler counterexample with the same properties was given in [93, Sec. 2.3]). One can find another (simpler) counterexample in [93] (see also the references therein). Furthermore, in [93] and [58], respectively, were constructed examples of left-invariant sub-Riemannian metrics with abnormal geodesics on some family of four-dimensional Lie groups and the six-dimensional Carnot group whose Lie algebra admits a basis {e1 , . . . , e6 } with the following nonzero brackets: [e1 , e2 ] = e3 ,

[e1 , e3 ] = e4 ,

[e2 , e3 ] = e5 ,

[e1 , e4 ] = e6 .

In the latter case, the distribution D has rank two and spans an orthonormal (left-invariant) basis {e1 , e2 }. Abnormal geodesics corresponding to the last two cases of a control system in the form (5.2) are exactly geodesics that have zero maximum M corresponding to the Pontryagin principle. 433

Let us give also a short description of the counterexample from [93, Sec. 9.5]. Let G be a fourdimensional Lie group whose Lie algebra L has two generating elements f and g such that (i) the elements f , g, [f, g], and [f, [f, g]] form a basis of the Lie algebra L; (ii) the element [g, [f, g]] is contained in the linear span of the elements f , g, and [f, g]; (iii) the element [g, [f, g]] is not contained in the linear span of the elements f and [f, g]. For example, it is possible to take the Lie group G = SO(3) × R and the set g = (k1 + k2 ) ⊕ 2, f = k1 ⊕ 1, where {k1 , k2 , k3 } is a basis of the Lie algebra so(3) of the Lie group SO(3) such that [k1 , k2 ] = k3 ,

[k2 , k3 ] = k1 ,

[k3 , k1 ] = k2 .

Let D be the subbundle in T G spanned by f and g. Define a left-invariant sub-Riemannian metric ρ on G assuming that f and g form an orthonormal basis of (sections) of the distribution D. By [93, Theorem 6], integral curves of the left-invariant vector field g are (strictly) abnormal locally (unique) shortest arcs of the sub-Riemannian space (G, ρ). In particular, it is easy to deduce from the last statement that in the general case it is impossible to find all geodesics and the precise shape of spheres in homogeneous sub-Riemannian manifolds of (topological) dimension n ≥ 4 using the methods proposed in [69, 117, 129]. On the other hand, these methods are applicable for sub-Riemannian manifolds defined by a regular distribution with nonholonomy degree 2 (see Definition 12), since in this case [117, Corollary 6.2] is valid (see correction to the paper [117]). In particular, these methods are valid in the case of homogeneous sub-Riemannian manifolds of (topological) dimension 3. In [27] V. N. Berestovskii and I. A. Zubareva found geodesics and exact formulas for spheres for a special left-invariant sub-Riemannian metrics on the three-dimensional Lie groups H 3 , SO(3), and SL2 R. The metrics considered in [27] have independent geometric significance since they are naturally defined by geometric constructions. Left-invariant (horizontal) totally nonholonomic distributions D on the corresponding Lie groups that allow one to define metrics represent, in all cases, horizontal distributions in principal bundles with one-dimensional (and hence commutative) Lie groups as fibers over planes with constant (respectively zero, positive, or negative) Gaussian curvature. In the second and third cases, the distribution D is the horizontal distribution of the Levi-Chivita connection on the bundle of unit vectors over the base. For all three cases, the projection p onto the base of the bundle is a submetry, while the shortest arcs are horizontal lifts of curves that are solutions of the classical isoperimetric Dido problem on the base. This allows one to find geodesics on groups without the Pontryagin maximum principle or Euler–Lagrange or Hamilton–Jacobi equations. Moreover, in the first case the projection p is an epimorphism of the Lie group H 3 onto the (vector) group V2 of parallel translations of the Euclidean plane. Spheres in groups are surfaces of revolution with two conical singularities on the rotation axis in the natural “first-type” canonical coordinates defined by the exponential mapping; this reflects the presence of the corresponding symmetries in metrized homogeneous manifolds under consideration. The problem on the shape of spheres for left-invaraint sub-Riemannian metric on the simplest Carnot group (i.e., the three-dimensional Heisenberg group H 3 ) has been solved. Earlier in [129], geodesics of sub-Riemannian metric on these three groups were found by using the Euler–Lagrange equations and more complicated shapes of spheres in H 3 in other coordinates (see below) were obtained. Similarly, for any left-invariant nonholonomic Finsler metric ρ on H 3 , the projection p is a submetry of the space (H 3 , ρ) onto the vector space V2 with a norm  · , while shortest arcs in (H 3 , ρ) are horizontal lifts of curves that are solutions of isoperimetric Dido problem for (V2 ,  · ). Based on this, G. A. Noskov in [107] found geodesics in (H 3 , ρ) purely geometrically without using the Pontryagin maximum principle. Let us examine the case of the Lie group H 3 more carefully. For any left-invariant sub-Riemannian metric d on H 3 , there exists an orthonormal basis {X, Y } of left-invariant vector fields that are 434

tangent to the corresponding totally nonholonomic distribution D. Moreover, the vector fields X, Y , and Z = [X, Y ] form a basis of the two-step nilpotent Lie algebra LH 3 . It follows immediately that any two sub-Riemannian manifolds of the form (H 3 , d) are isometric to each other. Therefore, we may assume that X = E12 , Y = E23 , Z = E13 , where Eij denotes the (3× 3)-matrix with unit in the ith row and jth column and zero in other entries. In [27], the coordinates are introduced as follows: ⎛ xy ⎞ 1 x z+ 2 ⎟ ⎜ 3 (6.1) exp(xX + yY + zZ) = ⎝0 1 y ⎠ → (x, y, z) ∈ R . 0 0 1 In these coordinates, the distribution D is given as the zero set of the contact 1-form 1 (6.2) ω = dz − (x dy − y dx). 2 For the Euclidean metric in these coordinates, this distribution can be described as the field of 2planes in R3 that are orthogonal to the velocity field of the one-parameter group of screw motions with the axis z and unit speeds of the translation motion along the z-axis and rotation about it. Therefore, the distribution D is closely related to the screw theory (see [84] and also the book [87] of A. P. Kotelnikov). In this case, the vector fields X, Y , and Z take the form X=

y ∂ ∂ − , ∂x 2 ∂z

Y =

∂ x ∂ + , ∂y 2 ∂z

Z=

∂ . ∂z

(6.3)

From (6.2) we immediately obtain the equality dω = −dx ∧ dy and the fact that the shortest arcs in (H 3 , d) are horizontal lifts of curves that are solutions of the classical isoperimetric Dido problem on the Euclidean base (of the projection p : R3 → R2 ; p(x, y, z) = (x, y)). In [129] the coordinates (x = x1 , y = x2 , and z = x3 ) on H 3 are introduced in such a way that the same vector fields X, Y , and Z on H 3 take the form ∂ ∂ ∂ ∂ , Y = +x , Z = . (6.4) X= ∂x ∂y ∂z ∂z In [129, p. 73], the coordinates on H 3 are introduced by the rule ⎛ ⎞ 1 x z − xy ⎝0 1 y ⎠ → (x, y, z). 0 0 1 Calculations show that in this case relations (6.3) do not hold, and we must introduce coordinates by the rule ⎛ ⎞ 1 x z exp(yY ) exp(xX) exp(zZ) = ⎝0 1 y ⎠ → (x, y, z). (6.5) 0 0 1 It follows from (6.1) and (6.5) that the passage from the coordinates in [27] to the coordinates in [129] is given by the rule  xy  . (x, y, z) → x, y, z + 2 Substantially about this, but with a mistake (and, naturally, without mentioning the later paper [27]), it have been said in [129, Remark, p. 68]; instead of the matrix considered there, the matrix from (6.5) 435

must be here. Under this passage, one actually gets “spheres with ears,” but nevertheless not with such “steep ears” as in [129, Fig. 5]. The coordinates (x, y, z) on H 3 defined by formula (6.5) reveal another relation (in addition to that mentioned in Sec. 5) of the distribution D to thermodynamics. It follows from Eqs. (6.4) that the distribution D in these coordinates is given by the formula dz − x dy = 0

⇐⇒

dz = x dy.

(6.6)

Performing the linear change of variables x = −P,

y = V,

z = W,

we obtain the formula (W  =)dW = −P dV,

(6.7)

which defines in thermodynamics the 1-form of work dW in the state spaces with two variables (V, P ) (for example, the volume and the pressure of a gas). Further information from thermodynamics for a standard volume-pressure system and about C. Caratheodory’s paper [40] is given here following [70]. One of the main purposes of [70] is an interpretation of results of S. Carnot’s work [42] mentioned above in the language of differential forms. The work itself contains no differential-form arguments and consists of statements expressed by exact language. The central statement of thermodynamics is that the work W and the heat Q are not scalar functions of a state but are varying magnitudes depending on passed paths in the state space. Their variations are naturally expressed by 1-forms W  and Q . The situation with variables (x, y, z) is similar if one admits only movements along paths that are tangent to the distribution D. On the language of differential forms, the first and second thermodynamics laws and the Carnot formula have the form (the first law), (6.8) dW  + dQ = 0   Q d =0 (the second law), (6.9) T 1 dW  = Q ∧ dT (the Carnot formula), (6.10) T where the scalar function T of (V, P ) is the absolute temperature. Any of these three formulas is a consequence of the other two. If the state space is simply connected, then the first law is equivalent to the existence of a scalar function E of (V, P ) (energy) such that W  + Q = dE. It is the law of conservation of energy. Similarly, the second law is equivalent to the existence of a scalar function S of (V, P ) (entropy) such that Q /T = dS. The Carnot formula (6.10) is a consequence of Carnot’s main principle (see [42, p. 20]). To formulate the Carnot principle, one needs the notions of isotherm, adiabatic curve, and Carnot cycle. An isotherm (respectively, an adiabatic curve) is a line on the state diagram that represents a thermodynamic process (in a macroscopic system) that occurs at a constant temperature (respectively, in which the system does not receive or lose heat). Isotherms of an ideal gas of constant mass satisfy equations of the form P V = const (the Boyle–Mariotte law ). Adiabatic curves are solutions of the ordinary differential equation Q = 0. A Carnot cycle is a path arranged as follows: isotherm → adiabatic curve → isotherm → adiabatic curve. The Carnot principle states that the work done by a system in a Carnot cycle is proportional to the emitted heat with a universal (independent of the system) proportionality constant, which depends only on the temperatures of two isotherms of the cycle. Note that in view of Green’s formula and 436

formula (6.7), this work is equal to the oriented area of the region (with volume element dV ∧ dP ) bounded by the Carnot cycle. In [40], C. Caratheodory gave an alternative form of the second thermodynamics law on the language of differential forms, which is appropriate for higher-dimensional state spaces. His argument includes two parts. In the first part, it is shown that Q = f dS, where f is a function, about which it is not yet known if it depends only on the temperature T . In the simply connected case, a necessary and sufficient condition for Q = f dS is the Frobenius condition Q ∧ dQ = 0. This condition automatically holds in dimension two, so this proof is necessary only for higher dimensions. Assuming the converse, i.e., Q ∧ dQ = 0, Caratheodory noted that any sufficiently close state can be attained by an adiabatic process. This attainability contradicts the statement “in a neighborhood of any state, there exist states that are unattainable from this state by adiabatic processes.” If we accept the last statement (Caratheodory’s formulation of the second thermodynamics law), then, reversing the argument, we get Q = f dS. In the second part of the argument, it is shown that f depends only on the temperature (and then, using the same freedom of argument that Clausius used, it may be defined as the temperature itself). The function S is the entropy of corresponding state space. Examples considered above naturally lead to the following generalization. Let p : M → B be a principal or associated (not necessarily vector) bundle with some invariant connection ∇, whose horizontal distribution D is totally nonholonomic. Then any Riemannian (or Finsler) metric ρ on the base B naturally assigns nonholonomic Riemannian (or Finsler) metric dc defined by the distribution D and the condition that the projection p : (M, dc ) → (B, ρ) is a submetry. Another equivalent definition is as follows: for x, y ∈ M , dc (x, y) is the greatest lower bound of lengths of piecewise smooth curves in (B, ρ), whose horizontal lifts in M join points x and y. Every shortest arc in (M, dc ), joining points x and y, is a horizontal lift of a solution to the so-called isoholonomic problem: find a path with minimal length in (B, ρ) whose horizontal lift joins points x and y. Isoholonomic variational problems for the length functional and energy of Riemannian metrics ρ have been considered in [53, 101]. In the paper [130] by A. M. Vershik and O. A. Granichina, solutions of isoholonomic variational problems for connections in principal bundles are related to solutions of generalized isoperimetric problems in the sense of A. D. Ioffe and V. M. Tikhomirov [80] on the base of bundle. It is proved that in many cases isoholonomic problems and the corresponding isoperimetric problems are equivalent. This interpretation is applied to the search for geodesics of some special left-invariant nonholonomic Riemannian and Finsler metrics on the Heisenberg group H 3 and the Engel group (recall that the Engel group is a (unique) connected, simply connected, nilpotent Lie group of dimension 4 and nilpotence degree 3); both these groups are Carnot groups. In the case of sub-Riemannian metrics on these groups, the Euler–Lagrange equations for geodesics can be integrated in elliptic functions. Since in the general case in [130] the Euler–Lagrange equations were used, in the evaluation of general results of this paper one must take into account results of [58, 93, 100, 103]. Let us cite the abstract of [103]: “Graded nilpotent Lie groups, or Carnot groups, play in sub-Riemannian geometry the same role as Euclidean spaces in Riemannian geometry. They are metric tangent cones for this geometry. Hoping that there exists a greater analogy between sub-Riemannian and Riemannian geometries, one might conjecture that sub-Riemannian geodesic flows on any Carnot group are completely integrable. We prove this conjecture to be false by showing that a sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of upper triangular (4 × 4)-matrices. As a consequence, we prove that the centralizer for the corresponding quadratic “quantum” Hamiltonian in the universal enveloping algebra of this Lie algebra is as small as possible.” 437

A Sasaki metric of a Riemannian manifold (M, g) is a Riemannian metric g1 on T M from [29, Sec. 1K]. One can characterize g1 by the following three properties: (1) the metric on any tangent vector space Mx ⊂ T M , x ∈ M , induced by metric g1 , coincides with the natural Riemannian metric on the Euclidean vector space (Mx , g(x)); (2) the natural projection p : (T M, g1 ) → (M, g) is a Riemannian submersion; (3) the length of each parallel vector field along any geodesic in (M, g), considered as a curve in (T M, g1 ), is equal to the length of this geodesic in (M, g). The following problem is a direct generalization of problems for groups SO(3) and SL2 R considered above and a particular case of isoholonomic problems for principal and associated bundles. Problem 4. Find geodesics on a homogeneous sub-Riemannian manifold, which is the bundle T 1 M of unit tangent vectors over a simply connected non-Euclidean symmetric space M of rank 1 (see [29, 72, 137]) with a sub-Riemannian metric defined by a (Riemannian) Sasaki metric on T 1 M and a horizontal distribution D of the Levi-Civita connection ∇ of the manifold M on T 1 M . It is proved successively that any locally compact two-point homogeneous (see Remark 1) space with inner metric is (a) homogeneous Riemannian manifold; (b) symmetric space; (c) Euclidean space or symmetric space of rank one (see [137]). Conversely, any Euclidean space or symmetric space of rank 1 is a (locally compact) two-point homogeneous space [137]. The history of the proof of these statements is briefly presented in [137]. Note that the statement (a) and the fact that any locally compact three-point homogeneous space with inner metric is a homogeneous Riemannian manifold with constant sectional curvature (and consequently n-point homogeneous in the simply connected case, see Remark 1, for all finite natural numbers n), is not difficult to deduce from results of the first author in [13–15]. The traditional proof of the fact that a two-point homogeneous Riemannian manifold is symmetric is based on the Montgomery–Samelson classification of Lie groups that act transitively on spheres (see [105]). The paper [121] of Z. Szabo gives a positive answer to the question of J. Wolf in [137]: Is it possible to prove, without using classification theorems, that two-point homogeneous Riemannian spaces are symmetric? To the subjects of this and previous sections are closely related the book of P. A. Griffiths [63], the survey of R. N. Shcherbakov [115] about works of Tomsk geometers in nonholonomic geometry, modern presentation of the main paper of V. V. Vagner on nonholonomic geometry in the paper of E. M. Gorbatenko [59], the survey of W. Liu and H. J. Sussman [93], and the book of R. Montgomery [102], while to the survey [93], singularity theory, and other areas of mathematics and mechanics is related the work [104] of R. Montgomery and M. Zhitomirskii. 7. Characterization of Invariant Totally Nonholonomic Distributions and Homogeneous Manifolds with Integrable Invariant Distributions Our next goal is to characterize G-invariant totally nonholonomic distributions D on G/H for a connected Lie group G and a compact subgroup H of G, and homogenous manifolds G/H, on which any G-invariant geodesic metric is a Finsler metric. Since the group H is compact, the group G admits an (auxiliary) Riemannian metric (·, ·), which is invariant under all shifts of the form lg , g ∈ G, and rh , h ∈ H. Then the tangent bundle T G is uniquely represented as the direct orthogonal (with respect to the metric (·, ·)) sum T G = C1 ⊕ C2 of distributions C1 and C2 that are invariant under the same shifts. The distributions C1 and C2 admit global orthonormal (with respect to (·, ·)) bases Y1 , . . . , Yn−l and Z1 , . . . , Zl of fields in LG, which, in contrast to the distributions C1 and C2 , are usually not invariant under shifts rh , h ∈ H. Lemma 1. The distributions C1 and C2 admit local bases that are orthonormal with respect to (·, ·) and invariant under all shifts rh , h ∈ H. 438

Proof. First, for any smooth vector field X on an open subset U ⊂ G/H, there exists a unique smooth ˜ on p−1 (U ), which is tangent to the distribution C1 and p-connected to X, i.e., vector vector field X such that ˜ X ◦ p = dp ◦ X. ˜ is said to be fundamental. It is clear that any fundamental vector field is Every such vector field X invariant under all shifts rh , h ∈ H. It is known that the projection p : G → G/H is a locally trivial (principal) bundle (with the fiber H) and there is a unique G-invariant Riemannian metric (·, ·)1 on G/H such that the mapping     p : G, (·, ·) → G/H, (·, ·)1

is a Riemannian submersion or, which is equivalent, a submetry with respect to the corresponding homogeneous geodesic metrics on G and G/H. It is also known that if U is an open subset in G/H contractible to a point in itself, then (1) there exists a smooth section s : U → G, i.e., a smooth mapping such that p ◦ s = idU ; (2) there exists an orthonormal with respect to (·, ·)1 local basis X1 , . . . , Xn−l of the distribution T (G/H) on U (see [79]). ˜n−l form a required local, orthonormal ˜1, . . . , X Then the corresponding fundamental vector fields X −1 with respect to (·, ·) basis of the distribution C1 on p (U ). The vector fields Z˜1 , . . . , Z˜l on p−1 (U ), required for the distribution C2 , are determined uniquely by the vector fields Z1 , . . . , Zl and the condition     Z˜i (g) = d r(s(p(g)))−1 g Zi s(p(g)) , i = 1, . . . , l, if g ∈ p−1 (U ). Theorem 16.

(1) The formulas C = dp−1 (D),

(7.1)

D = dp(C)

(7.2)

define a one-to-one correspondence between G-invariant distributions D on G/H and distributions C on G, including distribution C2 and invariant under all shifts lg , g ∈ G, rh , h ∈ H. (2) Any such distribution C is defined by formula C(g) = d(lg )(c), where c = C(e) is an arbitrary Ad(H)-invariant vector subspace in Te G = LG containing Te H = LH. (3) In this case, the distribution D is totally nonholonomic on G/H (respectively, integrable) if and only if the distribution C is totally nonholonomic on G (respectively, integrable). (4) The distribution C is totally nonholonomic on G (respectively, integrable) if and only if the least Lie subalgebra in Lie algebra LG containing c coincides with LG (respectively, c is Lie subalgebra in LG). (5) an Ad(H)-invariant vector subspace c satisfies the inclusion [LH, c] ⊂ c. (6) The last inclusion is also a sufficient condition if the Lie group H is connected, in particular, if the homogeneous manifold G/H is simply connected. Proof. Let D be a G-invariant distribution of rank k on G/H, where 0 ≤ k ≤ m = n − l. Then formula (7.1) defines a distribution C of rank k + l on G. By the G-invariance of the distribution D, the distribution C is invariant under all shifts lg , g ∈ G. Since the preimage p−1 (x) of every point 439

x ∈ G/H is a subset of the form gH ⊂ G, where g is an element in p−1 (x), then p ◦ rh = p for every element h ∈ H. Hence        dp(gh) d(rh )(g) C(g) = dp ◦ d(rh ) (g) C(g)       = d p ◦ rh (g) C(g) = dp(g) C(g) = D(x) = dp(gh)(C(gh)); moreover, Consequently,

     dim d(rh )(g) C(g) = dim C(gh) = k + l.   C(gh) = d(rh )(g) C(g) ,

i.e. the distribution C is also invariant under all shifts rh , h ∈ H. It follows from formulas (4.2) and (7.1) that C2 ⊂ C. It is clear that the distribution D1 := C ∩ C1 on G is invariant under all shifts lg , g ∈ G, rh , h ∈ H. Moreover, as in the proof of Lemma 1, with any vector field X on G/H, a ˜ (tangent to distribution C1 ) is p-connected, and if X is tangent to unique fundamental vector field X ˜ the distribution D, then X is tangent to the distribution D1 := C ∩ C1 . Consequently, the formula dp(C) = dp(D1 ) = D is valid, and thus formula (7.2) is also valid. Let C be a distribution on G. Assume that C includes the distribution C2 and is invariant under all shifts lg , g ∈ G, rh , h ∈ H. Assume also that x ∈ G/H and g, g1 ∈ p−1 (x). Then there exists h ∈ H such that g1 = gh. By the invariance of the distribution C under all shifts rh , h ∈ H, we have        dp(g1 ) C(g1 ) = dp(gh) C(gh) = dp(gh) d(rh )(g) C(g)         = dp ◦ d(rh ) (g) C(g) = d(p ◦ rh )(g) C(g) = dp(g) C(g) . Moreover, C2 ⊂ C. Therefore, by formula (7.2), the distribution D on G/H is well defined. In addition, in view of the same inclusion C2 ⊂ C, formula (7.1) is valid. The G-invariance of the distribution D follows from the invariance of C under all shifts lg , g ∈ G. It is clear that the second statement is equivalent to the first. Similarly to Lemma 1, in each open set of the form p−1 (U ), where U is an open set in G/H contractible in itself to a point, the distributions D1 and C2 admit local, orthonormal with respect ˜1, . . . , X ˜ k and Z˜1 , . . . , Z˜l , whose elements to (·, ·), and invariant under all shifts rh , h ∈ H, bases X are p-connected with the vector fields X1 , . . . , Xk , respectively, which constitute a local, orthonormal relative to (·, ·)1 basis of the distribution D in the set U , and zero vector fields on U . Moreover, as is ˜ j ] are p-connected with the vector fields [Xi , Xj ], i, j = 1, . . . , k, ˜i , X known, the local vector fields [X and the same holds for brackets of any orders of these vector fields. As a consequence, dp is a ˜k } onto the Lie algebra ˜1 , . . . , X surjective homomorphism of the Lie algebra generated by the basis {X generated by the basis {X1 , . . . , Xk }. The third statement of the theorem follows from this assertion and Proposition 3, with account of the basis Z˜1 , . . . , Z˜l for the integrable distribution C2 . The fourth statement is obvious if we use a global, orthogonal with respect to (·, ·) basis of the distribution C consisting of left-invariant vector fields {Y1 , . . . , Yk ; Z1 , . . . , Zl } on the Lie group G. The fifth statement follows from the well-known formula     (7.3) Ad exp(tv) (u) = Exp t ad(v) (u); v ∈ LH, u ∈ c, t ∈ R, where ad(v)(u) = [v, u], exp : LH → H is the exponential mapping of the Lie algebra LH into the Lie group H, while Exp denotes the exponential of the corresponding linear endomorphism of vector space. If G/H is simply connected, then the connectedness of the Lie group G and the exact homotopic sequence for locally trivial bundle p : G → G/H (see [78]) imply the connectedness of the compact 440

Lie group H, and consequently, the surjectivity of the exponential mapping exp : LH → H (see [72]). Now the last statement of the theorem follows from the same formula (7.3). Theorem 17. Every G-invariant inner (geodesic) metric on the homogeneous space G/H of a connected Lie group G with a stationary compact subgroup H is a Finsler metric if and only if (A) every G-invariant distribution on G/H is integrable. This is equivalent to the following condition; (B) every Ad(H)-invariant vector subspace c in LG containing LH is a Lie algebra. If H is connected, in particular, if G/H is simply connected, then the Ad(H)-invariance of the space c is equivalent to the inclusion [LH, c] ⊂ c. Proof. By Theorem 11, any G-invariant inner (geodesic) metric ρ on G/H is defined by a pair (D, F ), where D is a G-invariant, totally nonholonomic distribution on G/H and F is a G-invariant norm on D. Therefore, any such metric ρ is a Finsler metric if and only if (C) any G-invariant, totally nonholonomic distribution D on G/H coincides with the tangent vector bundle T (G/H). Based on Theorem 16, we see that this is equivalent to the condition (D) if the least Lie subalgebra in the Lie algebra LG containing an Ad(H)-invariant vector subspace c in LG with the condition LH ⊂ c, coincides with LG, then c = LG. Obviously, condition (D) follows from condition (B). Assume that condition (B) does not hold. Then there is an Ad(H)-invariant vector subspace c in LG containing LH, which is not Lie algebra. Hence the least Lie subalgebra f in LG containing c does not coincide with c. It is known that     Ad(g) [u, v] = Ad(g)(u), Ad(g)(v) , g ∈ G, u, v ∈ LG. Therefore, f is Ad(H)-invariant. Let s be the orthogonal (with respect to (·, ·)) complement to f in LG = Te G. Then the vector subspaces s and c1 = c ⊕ s are Ad(H)-invariant; moreover, LH ⊂ c1 and the least Lie subalgebra in the Lie algebra LG containing c1 is equal to LG = c1 . Consequently, condition (D) is violated. This contradiction shows that conditions (B) and (D) are equivalent. Based on Theorem 16, we conclude that conditions (B) and (A) are equivalent. The last statement of this theorem follows from the last two statements of Theorem 16. Definition 13. A homogeneous manifold G/H of a connected Lie group G with a compact stabilizer H is called a homogeneous manifold with integrable invariant distributions if it satisfies any of the equivalent conditions (A) or (B) from Theorem 17. 8.

Strong Subalgebras

By Theorem 17, the homogeneous manifold G/H of a connected lie group G with connected compact stabilizer H and with integrable invariant distributions is defined by a purely algebraic condition on their Lie algebras g = LG and h = LH: if c is a vector subspace in g such that [h, c] ⊂ c, then [c, c] ⊂ c, i.e., c is a Lie subalgebra in g. This condition was found in [14]. In [60] it was called the Berestovskii condition. Definition 14 (see [60]). A Lie subalgebra h in a Lie algebra g is said to be strong if it satisfies the Berestovskii condition; moreover, the corresponding Lie subgroup (in a connected Lie group G with the Lie algebra g) is compact. The last additional condition is oriented to geometric applications, but under purely algebraic study one may ignore it. The sense of the “strength” for a subalgebra h is as follows: it “endows” any vector subspace in g that contains it and is invariant under it with an algebraic structure (i.e., the structure 441

of a Lie subalgebra). By the way, subalgebras that are not strong should not be called weak (the term “weak subalgebra” is used in several various senses). A connected compact subgroup H in a Lie group G is said to be strong if the Lie subalgebra h ⊂ g of this Lie subgroup is strong. One can also give a direct definition in terms of Lie groups and invariant distributions on them, not passing on to Lie subalgebras. Precisely, in view of Theorems 16 and 17, the Berestovskii condition is equivalent to the following: any distribution C on a Lie group G, invariant under the left action of the group G on itself, containing the distribution T H (generated by the tangent space to the stabilizer H), and invariant under the right action of the group H, is integrable. Definition 15 (see [60]). A Lie algebra that has at least one proper strong subalgebra is called an adult Lie algebra. This term can be explained by the fact that a strong subalgebra is something like a nucleus (an analog of nuclei in biological cells), and the presence of a mature nucleus indicates the “adulthood” of an object or system. 9.

First Steps in the Study of Strong Subalgebras

We begin our study of strong subalgebras using several examples given in [13, 14]. 9.1. If h = {0}, the Berestovskii condition for a Lie algebra g is valid if and only if any vector subspace in g is a Lie subalgebra. It is easy to see that the last condition is always satisfied for onedimensional (commutative) Lie algebras, while in higher dimensions it is equivalent to the condition that any two-dimensional vector subspace in g is a Lie subalgebra. In other words, for any two elements X and Y in g, the bracket [X, Y ] is a linear combination of elements X and Y . Lie algebras g of such kind have been described by J. Milnor in “special example” [98, Example 1.7]. He proved in [98] the following statements for them: (1) there exists a linear mapping l : g → R such that [X, Y ] = l(X)Y − l(Y )X,

X, Y ∈ g;

(9.1)

(2) the kernel of the linear mapping l is a maximal commutative ideal in g. As a consequence, depending on the cases l = 0 or l = 0, (up to isomorphism) all Lie algebras g have the following form: (i) Abelian Lie algebras; (ii) Lie algebras of the form Ln = R +φ Rn−1 , n ≥ 2, i.e., semi-direct sums prescribed by the homomorphisms φ : R → End(Rn−1 ) such that φ(1) = En−1 is the unit matrix. They are called also “book Lie algebras” because of the specific form of the orbits in the coadjoint representation: here regular orbits are open half-planes resembling pages of a book. We see that in any dimension n ≥ 2, there are exactly two Lie algebras having a trivial strong subalgebra. Remark 10. 1. The Lie algebras Ln are sometimes called hyperbolic since the connected Lie groups Gn corresponding to them and supplied with any left-invariant Riemannian metric are isometric to the Lobachevsky spaces Λn . J. Milnor proved in [98] that the sectional curvature of these metrics is always negative and constant. A more detailed discussion of this result will be presented in the conclusion of this section. Since any homogeneous Riemannian manifold of negative sectional curvature is simply 442

connected (see [137]), the Lie group Gn is simply connected and consequently is defined by its Lie algebra Ln uniquely up to isomorphism. 2. The Lie group Gn can be considered as generated by parallel translations of the space Rn−1 and its homotheties. Thus, one can naturally consider the group Gn as a subgroup of the group of affine transformations of the space Rn−1 . In other words, the Lie group Gn is isomorphic to the group of real (n × n)-matrices of the block form   aEn−1 b , (9.2) 0 1 where En−1 is the unit (n − 1) × (n − 1)-matrix, a is a positive number, b is a (n − 1)-vector-column, and 0 is the zero (n − 1)-vector-row. In the sequel, we use the group of real ((n + 1) × (n + 1))-matrices   aEn (b0)t (9.3) 0 1 isomorphic to Gn , where (b0)t is the transposed n-vector-row with any (n − 1)-vector-row b and zero 0. This group can be interpreted as a subgroup of the group of affine transformations of the space Rn . 3. The Lie algebra Ln ⊕ R, which is algebraically slightly different from Ln , no longer satisfies the Berestovskii condition for h = {0}. In general, the Berestovskii condition is not additive, i.e., it generally does not withstand the formation of direct sums. 9.2. If (G, H) is a Riemannian symmetric pair (i.e., G/H is a Riemannian symmetric space), then, as was noted in [14], the corresponding pair of Lie algebras (g, h) satisfies the Berestovskii condition (this is indeed the classical result from the theory of symmetric spaces). A symmetric pair for a simple Lie group G has compact (respectively, noncompact) type if the group G is compact (respectively, noncompact). In this case, symmetric pairs of noncompact type are characterized by the fact that the stabilizer subalgebra h is the maximal compact subalgebra. Remark 11. In [117] the notion of a sub-Riemannian symmetric space was introduced. Owing to our results, it seems to be a combination of two incompatible notions. But note that even such space is a homogeneous sub-Riemannian manifold; it can be represented as a homogeneous space G/H, where (G, H) is a symmetric pair. We shall not give corresponding definition (see [117, Definition 9.1]). However, we note the following. In [117] some examples and classification for three-dimensional case of such spaces were presented. In [31], a classification of all simply connected contact sub-Riemannian symmetric spaces was obtained. 9.3. Let M = G/H be a strictly isotropically irreducible homogeneous space, i.e., such that the natural representation of the Lie algebra h on the space g/h (identified with a tangent space to the manifold M ) is irreducible. Then it is easy to understand that the pair (g, h) (which is said to be isotropy irreducible) satisfies the Berestovskii condition. Such spaces are especially interesting for us. Later on, we shall see that also, in the general description of strong subalgebras, they will play a very important role. Consider the Levy decomposition G = S · R of the Lie group G, where R is the radical of the connected Lie group G (the maximal solvable connected normal subgroup in G), while S is its semisimple part (Levi factor). One can easily deduce from the strong isotropic irreducibility of the homogeneous space G/H R = {e}, i.e., G = S is a semisimple Lie group. Here are some specific examples of strictly isotropically irreducible homogeneous spaces: (1) all irreducible symmetric spaces; (2) (nonsymmetric) homogeneous spaces of orthogonal groups having the form SO(dim(H))/H, where H is any simple Lie group embedded into SO(dim(H)) by means of the adjoint representation. 443

There are many other series of strictly isotropy irreducible nonsymmetric homogeneous spaces, as well as many “exceptional” homogeneous spaces (among them are also homogeneous spaces of exceptional simple compact Lie groups). Let us formulate now some general properties of strong subalgebras. Proofs here are almost obvious. 1. If h is a strong subalgebra in a Lie algebra g and h ⊃ h is a subalgebra including it, then h is also a strong subalgebra in g. 2. If h is a strong subalgebra in a Lie algebra g and g  is a subalgebra in g including h, then h is also a strong subalgebra in g  . 3. A Lie subalgebra h of a Lie algebra g is a strong subalgebra if and only if it is a strong subalgebra in any maximal subalgebra of Lie algebra g containing h. Now let us formulate some natural questions, the answers to which are interesting in connection with the notion of a strong Lie subalgebra. All semisimple Lie algebras are adult (for example, for a noncompact simple Lie algebra, one can take the maximal compact subalgebra as a strong subalgebra). Problem 5. Which Lie algebras are adult? For solvable g, a partial answer is given below (see Sec. 14). Problem 6. Describe all pairs (g, h) with a strong Lie subalgebra h for a given Lie algebra g, i.e., all strong subalgebras in a given Lie algebra. This is a very difficult problem. A partial question is given below and in Sec. 13. Problem 7. What are maximal strong subalgebras in a given Lie algebra? The answer to this question is also given below (Secs. 12 and 13). Problem 8. What are minimal strong subalgebras in a given Lie algebra? In this case, the question is namely about the minimal, but not the least, subalgebras. Usually, there is no least strong subalgebra, i.e., contained in all other strong subalgebras of given Lie algebras. For example, if g is a simple noncompact Lie algebra, then only maximal compact subalgebras are strong in it. All such algebras are conjugate with each other, but their intersection can be trivial. Therefore, if here would be the least strong subalgebra, then it should be trivial. But Lie algebras having trivial strong subalgebras were described above (see Sec. 9.1); there is no simple Lie algebras among them. It is proved in [98, Sec. 5] that every left-invariant Riemannian metric μ on the Lie group Gn with the Lie algebra Ln , n ≥ 2, has a constant negative sectional curvature K = −l2 ,

(9.4)

where l is the norm of the above-mentioned linear mapping l : Ln → R on the Euclidean space (Ln , μ(e)). It is proved in [98, Theorem 5] that if a Lie algebra is not commutative or is not one of Lie algebras Ln , n ≥ 2, then the corresponding Lie group admits a left-invariant Riemannian metric, giving both positive and negative Ricci curvatures. As a consequence of the results presented in the previous paragraph, one can characterize noncommutative Lie algebras with trivial strong subalgebras by the condition that every left-invariant Riemannian metric on the corresponding Lie group has sectional or Ricci curvatures of the same sign. It is well known that in the commutative case these curvatures vanish (see [98]). J. Milnor obtained Eq. (9.4) by means of calculations based on the formulas for the sectional curvatures of left-invariant Riemannian metrics on Lie groups derived by himself (in the book [44] of 444

J. Cheeger and D. G. Ebin, more convenient invariant formulas are given). We show how it is possible to get formula (9.4) “conceptually,” without using specific formulas for sectional curvature. The following two general principles are valid: (α) If μ1 and μ2 are left-invariant metric tensors on a simply connected Lie group G with the Lie algebra L = Te G and there is an automorphism λ of the Lie algebra L such that μ2 (e) = μ1 (e) ◦ (λ × λ), then the Riemannian manifolds (G, μ1 ), (G, μ2 ) are isometric. (β) If the metric tensor on a Riemannian manifolds is multiplied by a positive constant β, then all sectional curvatures are multiplied by the number 1/β. Based on formula (9.1), we can easily prove the following statement. (γ) Let μ1 (e) = (·, ·) be a fixed scalar product on the Lie algebra Ln , n ≥ 2. Then a linear transformation λ : Ln → Ln is an automorphism of the Lie algebra Ln if and only if under the passage from the scalar product (·, ·) to the scalar product μ2 (e) = (·, ·) ◦ (λ × λ), the norm of the linear mapping l : Ln → R does not change. (δ) Obviously, if a scalar product (·, ·) on Ln is multiplied by a positive number β, then the square of the norm of a linear mapping l is multiplied by the number 1/β. The Poincar´e model of the space Λn with constant sectional curvature −1 in the half-space   n     n 2 2 x2n with metric ds = dxk (9.5) R + = { x1 , . . . , xn )  xn > 0 k=1

is well known. (ε) Let (·, ·) be a scalar product on Ln such that l = 1, {e1 , . . . , en } is an orthonormal basis in (Ln , (·, ·)) with the condition l(en ) = 1. Then the linear span of the first n − 1 vectors is a commutative ideal in Ln , and the mapping (Gn , μ) → (Rn+ , ds2 ) defined by the formula   n   xk ek → x1 , . . . , xn−1 , exp xn exp k=1

is an isometry if μ is an invariant metric on Gn with condition μ(e) = (·, ·). Note that Gn acts on (Rn+ , ds2 ) (from the left) simply transitively and isometrically by the formula   n     xk ek (y1 , . . . , yn ) = exp xn y1 , . . . , yn + x1 , . . . , xn−1 , 0 exp k=1

(cf. with the last group from Remark 10, item 2), which was used in the proof of statement (ε). Formula (9.4) follows from statements (α)–(ε). 10.

Isotropically Irreducible Homogeneous Riemannian Spaces

Recall that a homogeneous space G/H is effective (locally effective) if the set of elements Fix(σ) ⊂ G fixing all points of the space G/H under the action of the group G on G/H, defined by formula (4.1), is a trivial (respectively, discrete) subgroup of the Lie group G. This is equivalent to the fact that the maximal normal (in G) subgroup of the group H is trivial (respectively, discrete). In this section, we consider only effective homogeneous spaces. The classification of strictly isotropically irreducible homogeneous spaces G/H and the corresponding pairs (g, h) of Lie algebras is well known; it was obtained for the symmetric case in the classical works of E. Cartan, and for the nonsymmetric case independently in the works of J. A. Wolf [138] and O. V. Manturov [95]. In [138], J. Wolf obtained a classification, presented global forms of Lie 445

groups G and H, and described many geometric properties of isotropy irreducible spaces, for example, full isometry groups, holonomy groups, invariant almost complex structure, and quaternion structures. The classification in [95] is based on the representation theory; it is complete except for two spaces: Sp(2)/ Sp(1) = SO(5)/ SO(3), where Sp(1) is embedded by means of an irreducible complex 4-dimensional quaternion representation of the group Sp(1), and SO(16)/ Spin(9), where Spin(9) is embedded by means of the spinor representation. In [138], the spaces Sp(n)/(Sp(1) · SO(n)), n ≥ 3, and SO(4n)/(Sp(1) · Sp(n)), n ≥ 2, are omitted. Thus, the works [138] and [95] together give a full classification for the nonsymmetric case. The full classification was obtained by M. Kr¨ amer [88] (for the corresponding pairs (g, h) of Lie algebras, it was also given in [30] without indication of embedding method of algebra h into g). By this classification, pairs of Lie algebras corresponding to strictly isotropically irreducible spaces are divided into the following three classes: (1) irreducible symmetric pairs of Lie algebras (they have been studied long ago in detail); (2) five infinite series of pairs of compact Lie algebras (two for Lie algebras of type su and three for Lie algebras so); (3) 33 exceptional compact pairs. Namely, the calculation of these pairs was the main difficulty of the classification. In this case, the term “exceptional” does not mean that they involve representatives from the class of exceptional Lie algebras (of the type G2 , F4 , E6 , E7 , E8 ). For example, among exceptional pairs is the pair (su(16), so(10)) with appropriate embedding of so(10) into su(16). Note that one can prove that any noncompact, homogeneous, Riemannian, strictly isotropically irreducible manifold is a simply connected symmetric manifold (see [137, Theorem 8.13.1]). At the same time, a nonsymmetric, (compact), homogeneous strictly isotropically irreducible space G/H has a simple Lie group G (see [30]). If G is one of the exceptional groups, then one can extract the classification from the works of Borel and de Siebenthal [33] and Dynkin [48]. Therefore, the case of classical Lie groups, i.e., groups G = SO(n), SU(n), or Sp(n), presents the main difficulty in the papers of Manturov and Wolf. In [134], M. Wang and W. Ziller gave a direct proof of Wall’s statement on the one-to-one correspondence between compact simply connected irreducible symmetric spaces from one side and compact strictly isotropically irreducible spaces of classical groups from the other side, whose inverse is described in Theorem 18 below for nonsymmetric spaces. The correspondence has some exceptions: (1) among symmetric spaces (a) all real and quaternion Grassmanian manifolds, except for SO(n+4)/(SO(n) SO(4)), SO(n+ 2)/(SO(n) SO(2)), Sp(n + 1)/(Sp(n) Sp(1)), and Sp(3)/(Sp(2) Sp(1)); there is no strictly isotropy irreducible spaces corresponding to them; (b) complex Grassmanian manifolds SU(n + 2)/S(U (n)U (2)), i.e., all compact simply connected irreducible symmetric spaces, which are simultaneously both Hermite symmetric and quaternion symmetric spaces; to any such space corresponds two strictly isotropy irreducible spaces: nonsymmetric SU(2n)/(SU(n) SU(2)) and symmetric Sp(n)/U (n); (c) compact symmetric spaces with rank one of classical groups, to any of which a point corresponds; (d) the spaces SU(3)/ SO(3) and G2 / SO(4) to which the same nonsymmetric strictly isotropy irreducible space Sp(2)/ Sp(1) = SO(5)/ SO(3) corresponds; (2) one nonsymmetric strictly isotropy irreducible space (e) SO(7)/G2 , corresponding to no symmetric space. Simultaneously, in [134] a modern proof for the first Cartan classification of irreducible symmetric spaces based on the classification of their holonomy groups is given. 446

In the paper of E. Heintze and W. Ziller [71] a simpler proof of the following result from [134] is given. Theorem 18. Let H be a compact connected subgroup of a Lie group G = SO(n), SU(n), or Sp(n) such that G/H is isotropically irreducible, while (G, H) is not a symmetric pair. We set ⎧ H for G = SO(n), ⎪ ⎨ ˜ := H · U (1) for G = SU(n), H ⎪ ⎩ H · Sp(1) for G = Sp(n), ˜ on Rn , Cn = R2n , or H = R4n respectively. Then the natural induced representation of the group H is an isotropy representation of some compact, simply connected, irreducible symmetric space (the so-called s-representation), except for the case where (G, H) = (SO(7), G2 ). In view of this theorem, one can easily obtain a classification of strictly isotropically irreducible nonsymmetric quotient spaces of classical groups from the classification of compact simply connected irreducible symmetric spaces. In this case, one needs to exclude s-representations of symmetric spaces, indicated above in (a) and (c), as well as the irreducible symmetric spaces Sp(n)/U (n), SU(n)/ SO(n), and (SO(8)/Z2 )/(SO(5) SO(3)), corresponding to s-representations of the Grassmanian spaces SU(n + 2)/S(U (n)U (2)), SO(n + 2)/(SO(n) SO(2)), and Sp(3)/(Sp(2) Sp(1)), respectively. Note that the homogeneous space (SO(7), G2 ) with any invariant Riemannian metric is isometric to the seven-dimensional real projective space P 7 (R) of constant positive sectional curvature (see [71]). Let us give information for the case of exceptional Lie groups G, following [138]. Recall that the rank of a compact Lie group is, by definition, the dimension of its maximal torus subgroup. It is well known that a homogeneous compact manifold G/H has nonnegative Euler characteristic; it is positive if and only if the rank of group H coincides with the rank of the group G (see [77]). B. Kostant proved [86] that a simply connected space G/H of positive Euler characteristic with compact connected Lie group G is indecomposable into the direct product of homogeneous spaces if and only if the Lie group G is simple. Therefore, results of [33], where a classification of closed Lie subgroups of maximal rank in simple compact Lie groups is presented, also gives a classification of indecomposable, compact, simply connected homogeneous spaces G/H of positive Euler characteristic (see also [137]). Using another result of B. Kostant (see [137, Theorem 8.13.3]), Wolf proves that a (simply connected) homogeneous space G/H with a (connected) compact simple Lie group G and positive Euler characteristic (consequently, even-dimensional) is nonsymmetric, strictly isotropically irreducible if and only if the center of the group H is a cyclic group of order 3. After this, based on [33], Wolf obtained in [138, Theorem 2.1] that the total number of such spaces is six ; they are       G2 / SU(3), F4 / SU(3) · SU(3) , E6 /Z3 / SU(3) · SU(3) · SU(3) ,           E7 /Z2 / SU(3) · SU(6) /Z6 , E8 / SU(9)/Z3 , E8 / SU(3) × E6 /Z3 . Thus, there is no classical Lie groups G in this list, but every exceptional Lie group (or its quotient group) is present in this list as G. Further, based on [48, Theorem 14.1], Wolf in [138, Theorem 3.1] (Wolf ascribed the authorship of this theorem to E. B. Dynkin) presented the following list of nine nonsymmetric, strictly isotropically irreducible homogenous spaces G/H with zero Euler characteristic and exceptional Lie group G:     G2 / SU(2), F4 / SU(2) · G2 , E6 /SU (3), E6 /G2 , E6 / SU(3) · G2 ,       E7 / SU(3), E7 / G2 · Sp(3) , E7 / SU(2) · F4 , E8 / G2 · F4 . We see that any exceptional Lie group is present in this list as Lie group G. 447

M. Wang and W. Ziller in [133] obtained a complete classification of simply connected, homogeneous (nonstrictly) irreducible manifolds G/H. An important significance of isotropically irreducible homogeneous spaces for geometry consists of the fact that in the compact case, in view of the general result of D. D. Bleecker [32], their unique (up to similarity) invariant Riemannian metric is a critical point for any C 1 -functional on the space of Riemannian metrics with fixed volume. Furthermore, in some sense (see exact formulations in [32] or in the review of this paper), this property characterizes these spaces. Note that Einstein metrics, i.e., Riemannian metrics with constant Ricci curvature, on any compact smooth manifolds are characterized as critical points for the functional of integral scalar curvature on the set of Riemannian metrics with fixed volume (see [30]). One can read about some recent achievements in the theory of homogeneous Riemannian spaces, among them homogeneous Einstein manifolds, in the monograph of V. V. Balashchenko, Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskii [6]. All isotropically irreducible homogeneous spaces G/H enter into the class of homogeneous spaces with integrable invariant distributions. 11.

Compact Simply Connected Homogeneous Manifolds with Integrable Invariant Distributions

To formulate the results below, we need to recall some known notions and information on homogeneous Riemannian manifolds. Every homogeneous Riemannian manifold M is isometric to some homogeneous effective space G/H of a (connected) Lie group G with compact stabilizer H, supplied with some G-invariant Riemannian metric μ (if M is compact, then the Lie group G is also compact). Since the Lie group H is compact, the Lie group G admits a Riemannian metric η invariant under all shifts lg , g ∈ G, rh , h ∈ H, and such that the canonical projection p : (G, η) → (G/H/, μ) is a Riemannian submersion. A Riemannian metric μ on G/H is said to be normal if one can find a biinvariant, i.e., invariant under left and right shifts, Riemannian metric η on G with the same property. Note that a connected Lie group admits a biinvariant Riemannian metric if and only if its Lie algebra is the direct sum of compact semisimple and commutative Lie algebras. A homogeneous Riemannian manifold is said to be normal if it is isometric to some homogeneous space G/H with normal G-invariant Riemannian metric μ (see [28]). It is well known that any normal homogeneous Riemannian manifold has nonnegative sectional curvature, and consequently also nonnegative Ricci and (constant) scalar curvatures (see [30]). A space G/H is of normal type (by Berard-Bergery) if every G-invariant Riemannian metric μ on G/H is normal (see [10]). Theorem 19 (see [10]). A homogeneous effective space G/H of a connected compact Lie group G with closed stabilizer H admits a G-invariant Riemannian metric of negative and, therefore of any (constant) scalar curvature if and only if G/H is not a space of normal type. The Berger–Wallach classification (corrected by B. Wilking, see [5, 9, 28, 132, 136]) of homogeneous Riemannian manifolds of positive sectional curvature is well known. In [19], the following simple topological and algebraic characterizations of homogeneous manifolds that admit invariant Riemannian metrics of positive Ricci curvature were obtained. Theorem 20. A homogeneous effective space M = G/H of a connected Lie group G with compact stabilizer H admits a G-invariant Riemannian metric of positive Ricci curvature if and only if the space M is compact (and hence the group G is also compact) and has a finite fundamental group π1 (M ). Moreover, any normal G-invariant Riemannian metric on G/H can be taken as such metric. Based on Theorem 20 and [61, Chap. 2, Sec. 3, Corollary 4], the following theorem can be proved. 448

Theorem 21. A homogeneous effective space M = G/H of a connected Lie group G with compact stabilizer H admits a G-invariant Riemannian metric of positive Ricci curvature if and only if G is compact and the Levi subgroup S in G, i.e., the largest connected semisimple subgroup in G, is transitive on M . The latter condition is equivalent to the equality G = SH. Moreover, G = SC, where C is a connected central subgroup in G. It is also known that for a nontrivial subgroup C from Theorem 21, such space M is a nontrivial torus bundle with totally geodesic fibers with respect to any G-invariant Riemannian metric on M over a homogeneous space M1 (of the same type) of a connected semisimple Lie group. The next lemma and four theorems were proved in [17]. In [10] L. Berard-Bergery gave the following characterization for spaces of normal type. Let G/H be a homogeneous effective space of a connected compact Lie group G with closed stabilizer H and g and h be the Lie algebras of the Lie groups G and H. On g, a (positive definite) scalar product ·, · is given. Denote by m the orthogonal complement to h in g with respect to ·, ·; mi , i ∈ I, are mutually orthogonal with respect to ·, ·, invariant and irreducible with respect to Ad H vector subspaces in m, giving its decomposition into the direct sum. In this case, G/H is of normal type if and only if for any such decomposition, [mi , mj ] = 0 if i = j; i, j ∈ I. Based on this result, we can prove the following theorem. Theorem 22. Every G-invariant distribution on the space G/H of normal type by Berard-Bergery is integrable. Now let G/H be a compact, effective, simply connected homogeneous space of a connected Lie group G with compact stabilizer H and integrable invariant distributions. Then G is compact, while ˜ be the linear isotropy group H is a compact connected Lie group. Further, let o = H ∈ G/H, H ˜ corresponding to subgroup H and acting on the tangent space To (G/H). Then every H-invariant decomposition into the direct sum (11.1) To (G/H) = U ⊕ V of vector subspaces defines G-invariant totally transversal foliations FU and FV on G/H that are tangent to these subspaces (see definition in [123]). The following lemma and three theorems are proved consecutively. ˜ has no nonzero fixed vector in the tangent space To (G/H). Lemma 2. A linear isotropy group H Theorem 23. Let FU and FV be G-invariant, totally transversal foliations on G/H defined by decomposition (11.1). Then there exists a real-analytic diffeomorphism of the manifold G/H onto FU × FV that transforms leafs of the foliations FU and FV to the corresponding fibers of the product FU × FV . In particular, all leafs of the foliations FU and FV are compact and simply connected. Theorem 24. For any G-invariant metric tensor μ on M = G/H, the homogeneous Riemannian manifold (G/H, μ) is isometric to the direct metric product (G1 /H1 , μ1 ) × · · · × (Gk /Hk , μ) of compact, simply connected homogeneous (strictly) isotropically irreducible Riemannian spaces. Also, the following assertions are valid: (1) G ⊂ G1 ×· · ·×Gk and H = G∩(H1 ×· · ·×Hk ) are compact connected Lie groups and, moreover, the homomorphisms-projections γi : G ⊂ G1 × · · · × Gk → Gi ,

i = 1, . . . , k,

are Lie-group epimorphisms; in addition the Lie groups Gi and Hi , i = 1, . . . , k, are connected; (2) the embedding homomorphisms G ⊂ G1 × · · · × Gk ,

H ⊂ H1 × · · · × Hk 449

induce a real-analytic diffeomorphism of the manifold G/H onto the manifold G1 /H1 × · · · × Gk /Hk . Theorem 25. A compact, simply connected, effective homogeneous manifold G/H is of normal type by Berard-Bergery if and only if G/H is a space with integrable invariant distributions. Based on properties (1) and (2) from Theorem 24, it was proved in [19] by algebraic methods that, in fact, under conditions and notation of Theorem 24, G = G1 × · · · × Gk ,

H = H1 × · · · × Hk ;

this implies the following theorem. Theorem 26. Any compact, simply connected, homogeneous effective manifold G/H with integrable invariant distributions is isomorphic to the direct product of compact, simply connected, homogeneous, strictly isotropically irreducible spaces. Theorem 27. For any (compact), simply connected, effective homogeneous space G/H of a connected compact Lie group G with closed stabilizer H, the following conditions are equivalent: (1) all G-invariant distributions on G/H are integrable; (2) the homogeneous space G/H is isomorphic to the direct product of compact, simply connected, strictly isotropically irreducible spaces; (3) the space G/H is of normal type by Berard-Bergery (see [10]); (4) all G-invariant Riemannian metrics on G/H have positive Ricci curvature; (5) all G-invariant Riemannian metrics on G/H have positive scalar curvature. Proof. The equivalence of conditions (1) and (3) is proved in Theorem 25. Due to Theorem 26, condition (2) is a consequence of condition (1); the converse statement is obvious. By the definition of the space G/H of normal type by Berard-Bergery, every G-invariant Riemannian metric on G/H is normal. Then, due to the simply connectedness of the space G/H and Theorem 20, condition (4) is a consequence of condition (3). Obviously, condition (5) is a consequence of condition (4). Finally, based on Theorem 19, we see that condition (3) follows from condition (5). Thus, the equivalence of all conditions, hence the validity of Theorem 27, is established. Remark 12. All proofs from the works of the first author, cited in this section, use no classification result. Conditions (4) and (5) in Theorem 27 and examples considered above naturally lead to the following problem. Problem 9. Is it true that simply connected homogeneous spaces G/H of connected Lie groups G by compact stabilizers H with integrable invariant distributions, not containing simultaneously as direct factors compact and noncompact homogeneous spaces with integrable invariant distributions, are characterized by some rigidity property for all G-invariant Riemannian metrics on G/H, for example, of having Ricci or scalar curvatures with constant and fixed signs? A homogeneous Riemannian manifold (M = G/H, μ) is called a geodesic orbit space (shortly, GOspace) if any of its geodesics is the orbit of some one-parameter subgroup of the isometry group G. Every homogeneous space (M = G/H, μ) with normal Riemannian metric is a GO-space. In [4], D. V. Alekseevsky and Yu. G. Nikonorov obtained the following results. If a simply connected GOspace (M = G/H, μ) of positive Euler characteristic, with a simple compact connected Lie group G, is not normal, then M is one of the flag manifolds M1 = SO(2n + 1)/U (n) 450

or

M2 = Sp(n)/U (1) · Sp(n − 1),

where n ≥ 2 and μ is an invariant Riemannian metric on M for which the space (M = G/H, μ) is not normal; in this case, μ depends on two real parameters, and for n = 2, the manifolds M1 and M2 are isomorphic. The manifolds (M1 , μ) and (M2 , μ) are weakly symmetric spaces. By the results of [86], the simplicity condition for a Lie group G in [4] is equivalent to the indecomposability of homogeneous spaces under consideration. A homogeneous space G/H of an arbitrary connected Lie group G with compact stabilizer H is called a rigid GO-space if the homogeneous Riemannian manifold (G/H, μ) is isometric to some GOspace for every G-invariant Riemannian metric μ on G/H. It is clear that any Berard-Bergery space of normal type is a rigid GO-space. In view of results of [4, 86] and Theorem 27, homogeneous spaces SO(2n + 1)/U (n) and Sp(n)/U (1) · Sp(n − 1) are rigid GO-spaces, but they are not products of isotropically irreducible spaces and thus are not homogeneous spaces with integrable invariant distributions, or Berard-Bergery space of normal type. Problem 10. Is any (at least, simply connected) homogeneous space G/H of a connected Lie group G with compact stabilizer subgroup H and integrable invariant distributions a rigid GO-space? 12.

Strong Subalgebras in Semisimple Lie Algebras

Here we consider properties and description of strong subalgebras in semisimple Lie algebras. The cases of strong subalgebras in solvable and general Lie algebras will be considered in the next sections. Recall that any semisimple Lie algebra is decomposed into the direct sum of simple Lie algebras called its simple factors (which are uniquely determined by the initial semisimple Lie algebra up to the order). If among these factors there is no compact Lie algebras, then g is called an algebra without compact factors. Theorem 28. Let g be a semisimple Lie algebra without compact factors. If h is a strong proper subalgebra in g, then h is a maximal compact subalgebra in g (and thus the pair (g, h) is symmetric of noncompact type). Proof. A strong subalgebra h is compact by condition. Let k ⊃ h be a maximal compact subalgebra of the Lie algebra g containing h. There is the Cartan decomposition g = k + m, where m is a vector subspace complement to k; moreover, [m, m] ⊂ k. Furthermore, since (g, k) is a symmetric pair, we have [m, m] = k (see the proof in [137] due to A. Borel, and also [109, p. 280, Exercise 2]). Consider the vector subspace v = h + m; it contains h and is invariant under the adjoint action of h on g, since [k, m] ⊂ m. Since h is a strong subalgebra by the condition, the subspace v must be a subalgebra in g. But then [m, m] ⊂ v and thus k = [m, m] is a subalgebra in v. As in Cartan decomposition k ∩ m = {0}, we have h = k, i.e., h is a maximal compact subalgebra in g. Because of Theorem 28, the question on strong subalgebras in semisimple Lie algebras without compact factors is exhausted (if we take into account the fact that a maximal compact subalgebra in a semisimple Lie algebra is always strong; see in Sec. 9.2 above). Let us turn to consideration of strong subalgebras in compact Lie algebras. Results that immediately imply the following general theorem were obtained in [18] (see also Theorem 27 above). Theorem 29. A subalgebra h in a compact semisimple Lie algebra g is strong if and only if there exist the decompositions into direct sums g = g1 ⊕ g2 ⊕ · · · ⊕ gn ,

h = h1 ⊕ h2 ⊕ · · · ⊕ hn ,

hi ⊂ gi ,

i = 1, 2, . . . , n,

such that all pairs (gi , hi ) are strongly isotropically irreducible. 451

Moreover, if a pair (gi , hi ) is not symmetric, then the Lie algebra gi is necessarily simple, but in the general case gi may be either simple or still the direct sum of two isomorphic simple Lie algebras. In the last case, we get symmetric pairs of the form (gi , hi ) = (k ⊕ k, diag(k)), where k is any compact simple Lie algebra. Remark 13. To any symmetric pair of the form (k ⊕ k, diag(k)) with a compact simple Lie algebra k, a compact irreducible symmetric space (K × K)/ diag(K) corresponds, where K is a compact connected (simple) Lie group with the Lie algebra k. Any such space with invariant Riemannian metric is isometric to the Lie group K with biinvariant, i.e., invariant under all left and right shifts, (inner) Riemannian metric ρ. On K, there is a unique (up to multiplication by a positive constant) biinvariant Riemannian metric ρ. The space (K, ρ) is Clifford–Wolf homogeneous in the following sense: for any two of its points, there exists a Clifford–Wolf translation (i.e., the isometry of the space (K, ρ), displacing all points of the space (K, ρ) one and the same distance), moving the first point to the second. As a Clifford–Wolf translation, one can take left or right shifts. It is proved in [23] that every simply connected Clifford–Wolf homogeneous Riemannian manifold is symmetric and isometric to the direct metric product of a Euclidean space of simply connected, simple Lie groups with biinvariant Riemannian metric and odd-dimensional Euclidean spheres of dimension ≥3 with induced inner metric (some mentioned factors may absent); the converse statement is obvious. The case of Lie groups (not necessarily compact or simple) with biinvariant Riemannian metric is separated from the general case by the existence of transitive group of Clifford–Wolf translations [11, 24], which in particular gives a negative answer to one conjecture of G. Busemann from [39]. Now let us consider the case of any semisimple Lie algebras. Let h be a strong subalgebra in some semisimple Lie algebra s (we do not assume that it is compact or has no compact factor). We have the decomposition s = sn ⊕ sc of the Lie algebra s into the direct sum of compact (sc ) and having no compact factor (sn ) of semisimple Lie algebras. Theorem 30. Let h be a strong subalgebra in a semisimple Lie algebra s = sn ⊕ sc . Then h = (h ∩ sn ) ⊕ (h ∩ sc ) is the direct sum of strong Lie subalgebras h ∩ sn and h ∩ sc in sn and sc , respectively. Proof. Consider the projection of h onto sn ; it is easy to see that it is a strong subalgebra in sn . But then this projection coincides with some maximal compact Lie subalgebra k in sn (due to Theorem 28). Let sn = k+m be the Cartan decomposition (here m is a standard, complement to k, vector subspace). Then h ⊂ k + sc . Let us set W = h + m; this vector subspace contains h and, obviously, is invariant under h (because m is the k-invariant complement to k). Since h is a strong subalgebra, we see that W is a Lie subalgebra in s. But, as was already noted above, [m, m] = k and thus the subalgebra k must be contained in h. This implies that h = k ⊕ (h ∩ sc ); moreover, k = h ∩ sn , and subalgebra h ∩ sc is, obviously, a strong subalgebra in sc . Thus Theorem 30 is proved. By Theorems 28, 29, and 30, the question on the description of strong subalgebras in semisimple Lie algebras is completely solved. Now consider strong subalgebras of small dimensions in compact simple Lie algebras. As was already noted, the zero subalgebra cannot be strong in a compact simple Lie algebra g. Furthermore, subalgebras of dimensions 1 and 2 also cannot be only abelian, because they are compact, while a nonabelian compact Lie algebra of the least dimension is a three-dimensional simple Lie algebra so(3) (or, equivalently, to su(2)). Therefore, such subalgebras also cannot be strong (because 452

of Theorem 29 and the description of isotropy irreducible pairs) in such g (besides a unique exception, one-dimensional strong subalgebra so(2) in the three-dimensional Lie algebra su(2)). Therefore, let us turn to the consideration of three-dimensional strong subalgebras in compact simple Lie algebras. Any such subalgebra should be nonabelian (by Theorem 29), and thus is isomorphic to the Lie algebra su(2) (or, equivalently, to the Lie algebra so(3)). Note that in classical Lie algebras such subalgebra is given by an appropriate linear representation φ of the Lie algebra su(2). Theorem 31. A subalgebra isomorphic to su(2) can be a strong subalgebra only in the following three simple compact Lie algebras: su(3) (symmetric case), sp(2)( so(5)), and the exceptional compact Lie algebra G2 (dimensions of these Lie algebras are equal to 8, 10, and 14, respectively). Embeddings of subalgebras are irreducible representations. Proof. By the above (or based on Theorem 22), it suffices to examine the list of strictly isotropy irreducible pairs (g, h) from [30] and separate pairs for which h = su(2). Note that three-dimensional simple strong Lie subalgebras are always minimal among strong Lie subalgebras. This follows from Theorem 29, since any proper subalgebra in su(2) is abelian and thus by Theorem 29 cannot be a strong subalgebra in a simple Lie algebra. Now consider strong subalgebras of dimensions 4 and 5. It follows from the classification of compact Lie algebras that there is only one (up to isomorphism) nonabelian compact Lie algebra of dimension 4: this is su(2) ⊕ R; there is also a unique compact nonabelian Lie algebra of dimension 5: this is su(2) ⊕ R2 . Both these algebras have nontrivial center. One can describe without especial labor, using the list from [30], all compact simple Lie algebras that contain strong subalgebras isomorphic to these Lie algebras of dimensions 4 and 5: this is only so(5) for the case of the Lie subalgebra su(2) ⊕ R (moreover, here we obtain symmetric pair). It is also possible to consider similarly the cases of strong subalgebras of dimensions 6 and 7, and etc. Remark 14. Isomorphic simply connected flag spaces Sp(2)/U (1) · Sp(1) and SO(5)/U (2) also correspond to the pair of Lie algebras (g, h) = (so(5), su(2) ⊕ R), but h is not a strong subalgebra in this case since these spaces are not spaces with integrable invariant distributions (see the end of Sec. 11). Consequently, different embeddings of the same Lie algebra into a given Lie algebra g may give both strong and not strong subalgebras. A similar statement is valid for homogeneous symmetric and isotropy irreducible spaces. 13.

Structure of Strong Subalgebras

In the previous seqtions, strong subalgebras in semisimple Lie algebras have been considered and described. In this section, we consider strong subalgebras in Lie algebras of general form, in particular, in solvable Lie algebras. In more detail, the case of solvable Lie algebras that contain strong subalgebras will be studied in the next section. Let g be a Lie algebra containing a strong subalgebra h. Consider the Levy decomposition g = s + r of the Lie algebra g (here r is the radical and s is the semisimple part; all such s are mutually conjugate). We denote by n the nilradical of the Lie algebra g. The Lie algebra h is compact; an appropriate Levy decomposition of g yields the decomposition h = hs + hr , where hr = h ∩ r is a torus in r (an abelian compact Lie algebra), while hs = s ∩ h is a compact Lie algebra (not necessarily semisimple). Note that, due to the compactness of a strong Lie subalgebra, it is always possible to find a Levy decomposition for a Lie algebra g containing it such that its semisimple part s is invariant under this Lie subalgebra. Further, the semisimple Lie algebra s is decomposed into the direct sum s = sn + sc of the compact sc and noncompact sn (more exactly, having no compact ideal) ideals. Recall that a pair (g, h) of Lie algebras is said to be effective if the subalgebra h contains no nontrivial ideals of the Lie algebra g. If a pair of Lie algebras is not effective, then, factorizing by 453

ideals contained in h, we obtain an effective pair. Similarly, on the level of Lie groups, one can pass from an arbitrary transitive action of a Lie group G to a locally effective action of the Lie group G with stabilizer H  , and also G/H = G /H  . To describe the structure of strong subalgebras h in the Lie algebra g, we need a series of lemmas. We always assume that h is a strong subalgebra in the Lie algebra g. Lemma 3. If pair (g, h) is effective, then h ∩ n = {0}. Proof. To simplify the proofs, we pass to the corresponding Lie groups, i.e., we consider a Lie group G whose Lie algebra is g, and a compact subgroup H in it corresponding to the subalgebra h (recall that, by our definition of a strong subalgebra, the corresponding Lie subgroup must be compact). One could also give the proof directly on the level of Lie algebras, but the passage to Lie groups here simplifies the arguments. Consider a subgroup H ∩ N in the nilradical N of the Lie group G. It is compact as the intersection of the compact subgroup H and the closed subgroup, nilradical N (which is always closed as the maximal connected nilpotent normal (Lie) subgroup). As is known, any compact subgroup in a connected nilpotent Lie group is central, thus H ∩ N is contained in the center Z(N ) of the Lie group N . Moreover, H ∩ N will also be central in G. In order to verify this assertion, consider some maximal compact subgroup K in G containing H. The intersection K ∩ N is a torus, which is a maximal compact subgroup in N . As is known, such a subgroup in a nilpotent Lie group N is unique; therefore, the action of the Lie group G by conjugations preserves this subgroup. But the group of automorphisms of the torus is discrete, while the Lie group G acting on this torus is connected, and thus K ∩ N is a central subgroup in G. But then also H ∩ N is a central subgroup in G; in particular, it is normal in G. The effectiveness of the pair (g, h) immediately implies the relation h ∩ n = {0}. Note that we used in this proof only the compactness of the Lie subalgebra h, but we do not need here the fact that it is a strong subalgebra. (However, we need it in the next lemma.) Lemma 4. If a pair (g, h) is effective, then the nilradical n of the Lie algebra g is abelian. Proof. As above, the subgroup H is compact. Consider the H-invariant vector complement W to [n, n] in the nilradical n; obviously, it is invariant under H. We obtain the decomposition n = [n, n] + W, where W is a subspace invariant under the adjoint action of the Lie algebra h. We set v = h + W ; this is a vector subspace invariant under h. Since h is a strong subalgebra, this subspace v is a subalgebra. However, it is known that the subspace complementary to [n, n] generates the nilpotent Lie algebra n (see, e.g., [36, Chap. I, Sec. 4, Exercise 4]). Therefore, n is contained in v = h + W . By Lemma 3, we have h ∩ n = {0}, while W ⊂ n, and hence W = n and thus [n, n] = {0}, i.e., the Lie algebra n is abelian. Lemma 5. The radical r of the Lie algebra g is decomposed into the semidirect sum r = a + b of two abelian Lie algebras a and b. Proof. By Lemma 4, the nilradical n of the Lie algebra g is abelian. Consider the h-invariant vector complement a1 ⊂ r to the subspace h ∩ r + n in r. We have r = a1 + (h ∩ r + n). The subspace h + a1 is h-invariant and, therefore, is a Lie subalgebra (moreover, its intersection a1 + h ∩ r with the radical r is abelian since [r, r] ⊂ n, as a consequence of the fact that a Lie algebra 454

is solvable if and only if its derived Lie algebra is nilpotent [36]), because h is a strong subalgebra. Now we set a = a1 + h ∩ r, b = n, and we obtain the required decomposition r = a + b of the radical r into the semidirect sum of two abelian Lie algebras. Let us turn to our main structural result. Theorem 32. Let h be a strong subalgebra in a Lie algebra g and the pair (g, h) be effective. Then (i) the subalgebra hn = h ∩ sn is a maximal compact subalgebra in the noncompact ideal sn of an appropriate semisimple part s of the Lie algebra g; (ii) the subalgebra hc = h ∩ sc is a strong subalgebra in the corresponding compact semisimple Lie algebra sc ; (iii) the subalgebra hs is the direct sum of hn and hc ; (iv) the radical r of theLie algebra g is decomposed into the semisimple sum a + n of an abelian subalgebra a (containing h ∩ r) and an abelian nilradical n. Proof. A compact Lie algebra h is decomposed into the direct sum of the central ideal h ∩ r and the subalgebra hs , which can be considered lying in some semisimple part s of the Lie algebra g. Therefore, h = hs + hr under an appropriate choice of the Levy factor s (here hs = h ∩ s and hr = h ∩ r). It is easy to see that the subalgebra hs is a strong subalgebra in the semisimple part s of the Lie algebra g (since the Lie subalgebra s commutes with hr ). Applying the method used above in the proof of Theorem 28, we see that hn is a maximal compact subalgebra in sn . Further, by Theorem 30, we have the decomposition hs = hn + hc , and the subalgebra hc is a strong subalgebra in the compact semisimple Lie algebra sc . Thus, items (i)–(iii) are proved. The statement (iv) was proved in Lemma 5. Consider some partial cases of Theorem 32. 1. If a Lie group G is simple and noncompact, then a subalgebra h is strong in g if and only if (g, h) is a symmetric pair. The same is valid also for any semisimple Lie algebra g without compact factors. Thus, we obtain a conversion of example from Sec. 9.2. 2. If a Lie group G is solvable, then G = A · N is the semidirect product of an abelian connected Lie group A and the abelian nilradical N . Every connected abelian Lie group is the direct product of a torus (a compact abelian Lie group) and a simply connected abelian Lie group. Therefore, A  Rk × T,



N  Rk × T  ,

where T and T  are tori (maybe trivial), and we may assume that h ⊂ T . For more details, see the next section. Now we consider some examples of strong subgroups in nonsemisimple Lie groups. 1. First, consider a semisimple product R = T · Rn of a simply connected normal subgroup Rn and a torus T ⊂ GLn (R) of linear transformations. Let us show that in the corresponding Lie algebra g, the subalgebra h = t (corresponding to the torus T ) is strong (and thus T is a strong subgroup in R). If v ⊃ h is an h-invariant vector subspace, then v = h + (k ∩ Rn ). Thus, the subspace v ∩ Rn is invariant under h, and therefore v, obviously, is a subalgebra. This proves that h is a strong subalgebra. Note that in the case under consideration, the Lie group R is solvable. 2. Consider a modification of the previous example. Let G = K · Rn be the semidirect product of a compact Lie group K and Rn (corresponding to the embedding K ⊂ GLn (R)). By the same 455

argument, as above in Example 1, we can prove that the subalgebra h = k is a strong subalgebra in the Lie algebra g of the Lie group G. Generally speaking, the Lie group G is not solvable. 3. Now we consider the semidirect product G = S · Rn , where S is a simple noncompact Lie group. We show that in the corresponding Lie algebra g, the maximal compact subalgebra h = k of the Lie algebra g is not necessarily strong. Let v ⊃ h be an h-invariant vector subspace in g distinct from h. It is known that since s is simple, k is not only a maximal compact subalgebra in s, but also a maximal subalgebra among all Lie subalgebras in s. Consequently, either v includes a simple Lie subalgebra, which is conjugate to the Lie algebra s, and thus one may assume that v ⊃ s, or v = k + v ∩ Rn . Consider in more detail the second case. The subspace v ∩ Rn is invariant under k, but it is not necessarily invariant also under s. For example, such deviation is possible even for s = sl2 (R). It is easy to verify that here the subalgebra k is strong in g if and only if any k-invariant vector subspace in Rn is also s-invariant. Simple Lie algebras s of such type exist, for example, all complex simple Lie algebras (since they are complexifications of their maximal compact subalgebras). Due to item (ii) of Theorem 32, some part of a strong algebra is situated in a compact semisimple Lie algebra. In an important partial case, we show how one can reduce the description of such subalgebras to the description of strong subalgebras in simple compact Lie algebras (see Sec. 12). This will be done in Lemma 6, but now we present some preliminary arguments. Consider again a homogeneous space G/H and an invariant inner metrics on it. In addition, assume that the stabilizer H is connected. In the general case, it has a finite number of connected components and the passage to the connected H can be performed by the passage to the finite-sheeted covering G/H0 → G/L over G/H (H0 is the connected component of the unit for the Lie subgroup H). A Lie group G does not necessarily coincide with the group I(G/H) of all isometries of the metric space G/H. Therefore, we can, if necessary, extend or narrow the isometry group G. Below, it will be convenient for us to narrow it and show that it is noncancellable. A transitive Lie group G is said to be noncancellable on G/H if its action is locally effective and there is no proper subgroups in G that are transitive on G/H. It is always possible to pass from any transitive Lie group to a noncancellable one using the passage to an appropriate connected transitive Lie subgroup and factorizing by an appropriate normal subgroup. Therefore, below in this section, in the study of strong Lie subgroups and subalgebras, we assume that the action of G on G/H is noncancellable. Let Sc = Πi Ki be the decomposition of the compact component Sc of the Levy factor S of a Lie group G into the almost direct product of simple compact Lie groups Ki . Denote by pi : sc → ki projections to direct summands for the corresponding Lie algebras. Lemma 6. If h is a strong subalgebra in sc , then (i) pi (h) are strong subalgebras in ki ; (ii) if the action of G on G/H is noncancellable, then pi (h) are proper strong subalgebras; (iii) if h is a maximal strong subalgebra, then it is the direct sum of strong subalgebras pi (h) ⊂ ki . Proof. Let vi ⊃ pi (h) be a vector subspace in ki invariant under pi (h). Consider p−1 i (vi ); it is an −1 h-invariant subspace in sc containing h. Since h is a strong subalgebra, pi (vi ) is a Lie subalgebra. Then its image under the projection pi , coinciding with vi , is also a Lie subalgebra. Consequently, the subspace vi is a subalgebra in ki and, therefore, pi (h) is a strong subalgebra in ki . This proves (i). We prove that under the condition of noncancellability for the action of the group G on G/H, the strong subalgebras pi (h) are proper, i.e., they cannot coincide with ki . Assume, for example, that 456

p1 (h) = k1 . Let us consider the subgroup G1 = Sn · (K2 · K3 · · · ) · R in the Lie group G, where Ki are the Lie subgroups in G corresponding to the subalgebras ki . It follows from the equality p1 (h) = k1 that G1 · H = G. But then it is obvious that the proper subgroup G1 ⊂ G is transitive on G/H; this contradicts the noncancellability of the action of the Lie group G. The contradiction obtained shows that all Lie subalgebras pi (h) are proper strong subalgebras in compact simple Lie algebras. This proves (ii).  In the general case, h is contained in the direct sum (pi (h)), but if it is maximal, then it must i coincide with this direct sum. This proves (iii). Note that Theorem 29 contains statements similar to Lemma 6, but in this theorem, direct summands into which a Lie algebra is decomposed is not always semisimple. Now we consider maximal strong subalgebras in Lie algebras of the general form (for subalgebras in semisimple Lie algebras, this question was considered in Sec. 12). We obtain the following statement. Theorem 33. Let the action of a Lie group G on G/H be noncancellable. Then any maximal connected strong subgroup H has the form Hc · Kn · T, where T is a maximal torus in the radical R of the Lie group G, Kn is a maximal compact subgroup in the component Sn of the Levy factor S for the Lie group G, and Hc is a maximal strong subgroup in the compact component Sc of the Levi factor S. In this case, Hc is the product of maximal strong subgroups in simple factors of the compact Lie group Sc . By Theorem 33, the description of maximal strong subalgebras in many respects is reduced to their description in compact simple Lie algebras, which has been completely performed above in Sec. 12, and in solvable Lie algebras, which will be discussed in the next section. In general, results of this section (in particular, Theorems 33 and 29) give a fairly detailed partial answer (based on the notion of a strong subalgebra and the Berestovskii condition) to the question from [14]: for which homogeneous spaces precisely are all inner metrics Finsler? Theorems 29 and 32 contain necessary conditions to the pair (g, h) in order that a Lie subalgebra h would be a strong subalgebra in g. Possibly, these conditions are close to sufficient; however, they will not be exactly sufficient (see example 3 after Theorem 32). 14.

Strong Subgroups in Solvable Lie Groups and Strict Abelian Subgroups

As was shown above, the description of strong subalgebras in many respects is reduced to considerations of strong subalgebras in semisimple and solvable Lie algebras. In this section, we prove several statements for the solvable case. We focus our attention not on strong subalgebras but on strong subgroups in Lie groups (since this is precisely useful on geometric applications that are interesting for us). Consider a solvable Lie group R = A · B, which is the semisimple product of two abelian Lie groups A and B. The radical of any Lie group containing a strong subgroup has this structure. In particular, if this Lie group is solvable, then it also has such structure. In this case, it can occur that both Lie groups A and B are not simply connected. However, the following statement holds. Proposition 4. If a solvable Lie group R has a strong subgroup, then it can be represented in the form of the direct product R = R1 × TB , where R1 = A · Rn is the semidirect product of an abelian Lie group A and a simply connected Lie group Rn and TB is a torus. 457

Proof. We set R = A · B,

A = TA × Rn ,

B = TB × Rm .

As was indicated above, h ⊂ tA . Consider the h-invariant vector complement in r to a maximal torus t (t = tA + tB ) of the Lie algebra r; r = W + t. Let us set v = h + W ; this is an h-invariant subspace, and since h is a strong subalgebra, this space is also a subalgebra. We have r = v + U + tB , where U is a subalgebra in tA complement to h. The subspace v + U contains h and is invariant under h; therefore, it is a subalgebra. Since tB ⊂ Z(r) (because b is the nilradical in r), we obtain the decomposition into the direct sum r = (v + U ) ⊕ tB . This decomposition yields the required decomposition into the direct product R = R1 × TB , where R1 is the connected Lie subgroup in R corresponding to the Lie subalgebra v + U . Note that in Proposition 4, the subgroup H is contained in the Lie group R1 with the simple connected nilradical, and it is a strong subgroup in R if and only if it is a strong subgroup in R1 . We also note that indeed in most cases TB = {e}, but this subject requires a separate consideration. As a result, we arrive at a consideration of strong subgroups in solvable Lie groups of the form R = A · B, semidirect products of abelian Lie groups; moreover, a Lie group B is simply connected, while H ⊂ A. But the problem of description of Lie groups R of the indicated form is reduced to the study of abelian subgroups in linear groups Gl(B), but subgroups possessing as will be shown below, very specific properties. Let R = A · B be the semidirect product of abelian Lie groups A and B, where B  Rn . As a semidirect product, the Lie group R is specified by a homomorphism φ : A → GLn (R). Let us set A = φ(A) ⊂ GLn (R),

H  = φ(H)

and consider the algebraic closure A  of the abelian Lie subgroup A in GLn (R). We have the Chevalley decomposition A  = As · Au of the abelian algebraic group A  into the direct product of totally reducible subgroup As and unipotent subgroup Au . Proposition 5. If H is a strong subgroup in the solvable Lie group R = A · B (moreover, B is simply connected), then Au = {e}, i.e., the subgroup A  is totally reducible. In particular, the group φ(A) is also a totally reducible linear group. For the proof we need one lemma. Lemma 7. Let R = A · B be the semidirect product of abelian Lie groups and B  Rn . If H is a strong subgroup in R, then any H  -invariant vector subspace W ⊂ B is also A -invariant. Proof. Let W ⊂ B be an H  -invariant vector subspace. Let us set v = a + W , where a is the Lie subalgebra corresponding to the Lie subgroup A; this is a vector subspace in r invariant under h, and hence it is a subalgebra. But then [a, W ] ⊂ W , i.e., the subspace W is also invariant under A . Proof of Proposition 5. Assume that Au =  {e}. Consider the subspace of invariants V1 = B Au corresponding to the group Au ; it is nonzero since the group Au is unipotent. Consider the H  -invariant 458

vector complement V2 to V1 in B, B = V1 ⊕ V2 . By Lemma 7, the subspace V2 is also A -invariant. Now we set V2 = (V2 )Au ; this is also a proper subspace in B. Further, this construction is repeated several times if necessary;  as a result we obtain the decomposition into the direct sum B = Vi of subspaces invariant under i

Au ; moreover, the restriction of the action of the group Au to every subspace Vi is the identity transformation. But then also the action of the group Au on B is trivial, i.e., Au = {e}. Thus, because of what has been proved above, R = A · B is the semidirect product of Abelian Lie groups (moreover, B  Rn ) corresponding to the homomorphism φ : A → Gln (R); moreover, the subgroup A is Abelian and totally reducible, and the same is also valid for the corresponding Lie algebra a ⊂ gln (R). But then a is contained in some Cartan subalgebra of the Lie algebra gln (R) (here a Cartan subalgebra is a maximal abelian Lie subalgebra consisting of totally reducible elements). But Cartan subalgebras in gln (R) are well known (see, e.g., [118]); they have the form ⎛ ⎞   ai −bi ⊕⎝ [cj ]⎠ , bi ai i

j

i.e., they are direct sums of several cells of order 2 of standard form and several cells of order 1, where ai , bi , and cj are arbitrary real numbers. The number k of cells of order 2 is one of the numbers !n + 1" ; it is equal to the dimension of the maximal torus in the Cartan subalgebra. Thus, B 0, 1, . . . 2 is decomposed into the direct sum B = U ⊕ V of two vector subspaces; furthermore, the restriction of A to U is contained in the direct sum of standard cells of second order, while the restriction to V consists of diagonal matrices. The stabilizer subalgebra, because of its compactness, is contained in the maximal torus of the Cartan subalgebra, in particular, in the direct sum of cells of second order. Its action on V is trivial. The following proposition shows which new restrictions are imposed on a solvable Lie group R by the presence of strong subgroup in it. Proposition 6. If a Lie group R = A · B is adult, then in some basis of the space B = Rn we have   ai −bi  ⊕ {REn−2k }, a ⊂ b i ai i

where En−2k is the identity matrix of order n − 2k. In other words, the restriction of the group A to the subspace V described above consists of scalar matrices. Proof. If the Lie algebra r is adult, then it has a strong Lie subalgebra h. Assume that the restriction of the action of the Lie subalgebra a to the subspace V is not scalar. Then in V there exists a vector v ∈ V such that the straight line v spanned by it is not invariant under this restriction. We set W = h + v; this is a subspace generated by the subalgebra h and the vector v. It is clear that W is invariant under the action of the subalgebra h (since the action of h on V is trivial). Therefore, by Lemma 7, W is invariant also under the action of a . But this is not so, since the straight line v is not invariant under this action. We arrive at a contradiction. Therefore, the restriction a to the subspace V is scalar. For k = 0, Proposition 6 gives the Milnor result, which was mentioned above (about Lie algebras with zero strong subalgebra). One can give a description in the case of Lie algebras for which k = 1 (i.e., Lie algebras for which the stabilizer h is one-dimensional); we leave this to the interested reader. 459

The following definition is motivated by Lemma 7. Definition 16. A connected abelian linear group A ⊂ GL(V ) is said to be strict if any T -invariant vector subspace in V (where T is a maximal torus in A) is also A-invariant. Theorem 34. Let G = S · R be a connected Lie group whose semisimple part S has no compact factors. This group is adult (i.e., has at least one strong subgroup) if and only if its radical R can be represented in the form R = (A · N ) × T, where T is a torus, N  Rn , and A = φ(A) is a strong subgroup. Proof. If the Lie group G is adult, then it contains, by definition, a strong subgroup H. Then the statement of Theorem 34 follows from the above results of this section. Conversely, if the group G has the form indicated in formulation of theorem, then a maximal compact subgroup in G can be taken as a strong subgroup. Acknowledgment. The first author was partially supported by the Russian Foundation for Basic Research (project Nos. 08-01-00067-a and 11-01-00081-a), the grants RFBR-BRFBR 10-01-90000-Bela, NSH-6613.2010.1, and the project “Quasiconformal Analysis and Geometrical Aspects of Operator Theory.” The second author was partially supported by the Russian Foundation for Basic Research (project No. 11-01-00465-a). REFERENCES 1. H. Abels and G. Margulis, “Coarsely geodesic metrics on reductive groups,” in: Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge (2004), pp. 163–183. 2. A. A. Agrachev and A. Marigo, “Rigid Carnot algebras: a classification,” J. Dynam. Control Syst., 11, No. 4, 449–494 (2005). 3. A. A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint, Encycl. Math. Sci., 87, Springer-Verlag, Berlin (2004). 4. D. V. Alekseevsky and Yu. G. Nikonorov, “Compact Riemannian manifolds with homogeneous geodesics,” SIGMA Symmetry Integrability Geom. Methods Appl., 5, p. 093 (2009). 5. S. Aloff and N. Wallach, “An infinite family of 7-manifolds admitting positively curved Riemannian structures,” Bull. Am. Math. Soc., 81, 93–97 (1975). 6. V. V. Balashchenko, Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskii, Homogeneous Spaces: Theory and Applications [in Russian], Poligraphist, Khanty-Mansiisk (2008). 7. C. B¨ar, Carnot–Caratheodory Metriken, Diplomarbeit, Bonn (1988). 8. A. Bendikov and L. Saloff-Coste, “On the absolute continuity of Gaussian measures on locally compact groups,” J. Theor. Probab., 14, No. 3, 887–898 (2001). 9. L. Berard-Bergery, “Les vari´et´es riemanniennes homog`enes simplement connexes de dimension impaire `a courbure strictement positive,” J. Math. Pure Appl., 55, 47–68 (1976). 10. L. Berard-Bergery, “Sur la courbure des m´etriques Riemanniennes invariantes des groupes de Lie ´ Norm. Super. 4 s´er., 11, 543–576 (1978). des espaces homog`enes,” Ann. Sci. Ec. 11. V. N. Berestovskii, “Homogeneous Busemann G-spaces,” Sib. Mat. Zh., 23, No. 2, 3–15 (1982). 12. V. N. Berestovskii, “Homogeneous spaces with inner metric,” Dokl. Akad. Nauk SSSR, 301, No. 2, 268–271 (1988). 13. V. N. Berestovskii, “Homogeneous manifolds with an intrinsic metric, I,” Sib. Mat. Zh., 29, No. 6, 17–29 (1988). 14. V. N. Berestovskii, “Homogeneous manifolds with an intrinsic metric, II,” Sib. Mat. Zh., 30, No. 2, 14–28 (1989). 460

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