Journal of Mathematical Sciences, Vol. 149, No. 1, 2008

BENDING OF SURFACES. III I. Ivanova-Karatopraklieva, P. E. Markov , and I. Kh. Sabitov

UDC 513

Abstract. A survey of works on discrete and continuous rigidity/nonrigidity and infinitesimal rigidity/nonrigidity of multidimensional surfaces, mainly in Euclidean spaces, is given. As a starting point for the methods of investigation, one considers three forms of the main theorem of the theory of surfaces (in local coordinates, in the invariant form, and in terms of exterior differential forms).

CONTENTS 9. Main Theorem of the Theory of Surfaces in Different Forms . . . . . . . . . . . . . . . . . . . 9.1. Classical Formulation of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. The Main Theorem in the Invariant Form . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Main Theorem in Terms of Exterior Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Some General Criteria of the Congruence and Unique Definiteness . . . . . . . . . . . . . 10. Isometries and Bendings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Global and Continuous Rigidity of Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 10.2. Global and Continuous Rigidity of Surfaces of Codimension p > 1 . . . . . . . . . . . . . 11. General Problems of the Theory of Infinitesimal and Analytic Bendings . . . . . . . . . . . . 11.1. Definitions of Infinitesimal and Analytic Bendings . . . . . . . . . . . . . . . . . . . . . . 11.2. Infinitesimal Motions and Trivial Infinitesimal Bendings . . . . . . . . . . . . . . . . . . 11.3. Rotation Fields and the Main System of Equations . . . . . . . . . . . . . . . . . . . . . 11.4. Projective Invariance of the Infinitesimal Rigidity of First Order . . . . . . . . . . . . . . 12. Theorems on Infinitesimal Rigidity and Nonrigidity . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Theorems on Infinitesimal Rigidity and Nonrigidity of Hypersurfaces . . . . . . . . . . . 12.2. Theorems on Infinitesimal Rigidity and Nonrigidity for Surfaces with Codimension p > 1 12.3. Infinitesimal Bendings of Fibered Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4. Rigidity Theorems for Multidimensional Surfaces with Boundaries . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

862 862 864 867 869 870 870 872 877 877 879 881 883 883 883 884 885 887 888

This review continues [66, 67]. According to the plan stated in [66], this paper is devoted to bendings and infinitesimal bendings of higher-dimensional surfaces. The review contains results related to hypersurfaces and surfaces of arbitrary finite codimension. Results of the theory of bendings and infinitesimal bendings of higher-dimensional surfaces can be divided into two classes with respect to applied methods of proof. Results of the first class are based on the study of the Gauss, Peterson–Codazzi, and Ricci equations and the use of the main theorem of the theory of surfaces. This approach can be applied only to surfaces in spaces of constant curvature. Results of the second class are based on the investigation of the Schl¨ afli equations in different forms and concern surfaces in Riemannian and pseudo-Riemannian spaces of arbitrary curvature. The first class of results is much wider than the second class. In this review, as a rule, we restrict ourselves to the study of surfaces in Euclidean spaces. The following part of the review will be devoted to the theory of bendings of surfaces in Riemannian spaces. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 3–56, 2006. c 2008 Springer Science+Business Media, Inc. 1072–3374/08/1491–0861


861

9. Main Theorem of the Theory of Surfaces in Different Forms In what follows, the word “manifold” means a smooth, Hausdorff, orientable manifold satisfying the second countability axiom. Without loss of generality, we can assume that manifolds are C ∞ -smooth. A Euclidean space is an affine space whose vector part is equipped with a positive-definite bilinear form called the scalar product. If we introduce an affine coordinate system in a Euclidean space, then the C ∞ -structure appears on this space and, therefore, it can be considered as a manifold. The tangent space at any point of a Euclidean space and the scalar product are canonically identified with the vector part of this space and the Riemannian metric, respectively. The Gauss, Peterson–Codazzi, and Ricci equations can be written in the following three forms: (1) the classical, local-coordinate form (Eisenhart [54], Schouten and Struik [123], Aminov [3]); (2) the invariant form (Szczarba [141, 142], Kobayashi and Nomizu [81]); (3) the form using differential forms (Yanenko [161–163], Griffiths, Bryant, Berger, and Yang [7, 19]). For each of these forms, there exists its own version of the main theorem. 9.1. Classical Formulation of the Main Theorem. Let X be an n-dimensional manifold and E be the m-dimensional Euclidean space, 2 ≤ n ≤ m. We consider a C r -immersion z : X → E, r ≥ 1. At any point x ∈ X, the value z(x) is identified with the radius-vector of the corresponding point in E with respect to some fixed orthonormal coordinate system in E. A surface in E is the image F = z(X) considered together with the immersion z. We denote by 1 x , . . . , xn the local coordinates of a point x ∈ X. Then the immersion z (and, therefore, the equation of the surface F ) can be written in the form z = z(x1 , . . . , xn ) in a neighborhood of any point of X, where z(x1 , . . . , xn ) is a vector-valued function of class C r . , i = 1, . . . , n, is called the tangent The subspace Tx F of the space E generated by vectors z,i (x) = ∂z(x) ∂xi plane of the surface F at the point x ∈ X (or at the point z(x) ∈ F ). The orthogonal complement Tx⊥ F to the tangent plane Tx F in E is called the normal plane of the surface F at the point x (or at the point z(x)). Usually, the tangent and normal planes are considered as vector spaces without an affine structure. The tangent plane is the image of the tangent space Tx (X) of the manifold X at the point x under the tangent mapping Tx z : Tx (X) → E. The number p = m − n = dim Tx⊥ F is called the codimension of the surface F . At any point, there exists an orthonormal basis (νσ )pσ=1 of the plane Tx⊥ F such that the corresponding vector fields νσ are fields of class C r−1 . The metric on X induced by the immersion z and the scalar product in E is denoted by I(z) or simply I. In local coordinates, it can be written in the form n
gij dxi ⊗ dxj I(z) = i,j=1

(see [18, 81]), where (9.1) gij = z,i ·z,j , and the dot denotes the scalar product in E (sometimes, we denote the scalar product in E by ∗, ∗ ). The covariant derivative in xi of a tensor field A on X with respect to the connection determined by the metric I(z) is denoted by A,i . If z is a C r -immersion, r ≥ 2, then we decompose the covariant derivatives z,ij and νσ ,i with respect to the basis (z,i ; νσ ) in E and thus obtain the Gauss and Weingarten formulas z,ij =

p
σ=1

bσij νσ ,

νσ ,i = −

n
k,l=1

g kl bσki z,l +

p
τ =1

µτσi ντ ,

(9.2)

where i, j = 1, . . . , n, σ = 1, . . . , p, g ij are the elements of the matrix inverse to (gij ), the coefficients bσij = z,ij ·νσ form a twice-covariant, symmetric tensor field with respect to the subscripts i and j, and the coefficients µτσi = ντ · νσ ,i form a covariant tensor field with respect to the subscript i. The bilinear 862

n 

bσij dxi ⊗dxj is called the second fundamental form of the surface F with respect n  to the normal νσ and the linear differential form κστ = µτσi dxi is called the torsion form of the normal

differential form IIσ =

i,j=1

i=1

νσ with respect to the normal ντ . The torsion forms κστ satisfy the condition κστ = −κτσ . If z is a C 3 -immersion, then the integrability conditions of the Gauss and Weingarten formulas (9.2) yield the Gauss, Peterson–Codazzi, and Ricci equation [54]: p


µτσi,j − µτσj,i

 bσik bσjl − bσil bσjk = Rijkl , p σ=1
 τ σ  σ σ bik µτ j − bτij µστk , bij,k − bik,j = τ =1 p  n 
 τ σ 
kl τ σ µτρi µρσj − µτρj µρσi , = g bkj bil − bki bjl +

(9.3a) (9.3b) (9.3c)

ρ=1

k,l=1

where Rijkl is the curvature tensor of the metric I(z), which can be calculated by the formulas   2 n
∂ 2 gjk ∂ 2 gjl ∂ gil 1 ∂ 2 gik + Rijkl = + − − g hs (Γjks Γilh − Γjls Γikh ) , 2 ∂xj ∂xk ∂xi ∂xl ∂xj ∂xl ∂xi ∂xk h,s=1   1 ∂gih ∂gkh ∂gik , Γikh = + − 2 ∂xk ∂xi ∂xh

(9.4)

i, j, k, l = 1, . . . , n, τ, σ = 1, . . . , p. For hypersurfaces (p = 1), the torsion forms vanishes, and the Gauss and Weingarten formulas become ν,i = −

z,ij = bij ν,

n


g kl bki z,l ,

(9.5)

k,l=1

where ν is the field of unit normals of the surface and bij are the coefficients of the second fundamental form II of the surface with respect to the normal ν. Equation (9.3c) (the Ricci equation) becomes an identity and Eqs. (9.3a) and (9.3b) take the form bik bjl − bil bjk = Rijkl ,

bij,k − bik,j = 0.

(9.6)

Now we present the classical statement of the main theorem (the Bonnet theorem) Theorem 9.1 (see [3,54]). Let X be an n-dimensional manifold on which the following objects are defined : n  gij dxi ⊗ dxj of class C r−1 , r ≥ 3, (i) a positive-definite bilinear differential form ds2 = i,j=1 n  bσij dxi ⊗ dxj of class C r−2 , and (ii) p bilinear differential forms B σ = (iii)

p(p−1) 2

i,j=1 n 

linear differential forms kστ =

i=1

µτσi dxi of class C r−2 ,

where kστ = −kτσ , τ, σ = 1, . . . , p. Assume that the coefficients of these forms satisfy Eqs. (9.3) and (9.4). Then any point x ∈ X possesses a neighborhood U such that there exists a C r -embedding z : U → E satisfying the following conditions: (1) the metric I(z) on U induced by the embedding z coincides with ds2 ; (2) at any point of U , there exists an orthonormal basis (νσ )pσ=1 in the normal plane of the surface z(U ) in E such that the second fundamental forms IIσ and the torsion forms κστ of the surface z(U ) with respect to the normals (νσ )pσ=1 coincide with the forms B σ and kστ , respectively. Such an embedding z is unique up to a transformation of the form P · z + ω,

(9.7) 863

where P is an arbitrary orthogonal (m × m)-matrix and ω is an arbitrary vector in E; the dot means the matrix product. This theorem is local, i.e., it guarantees that only a sufficiently small neighborhood on X can be embedded in E. As concerns the uniqueness, this implies the global version of the theorem. Papers [79, 120, 121] are devoted to the global existence theorem; the additional condition that the manifold X is simply connected allows one to state the global version of the existence theorem. If X is not simply connected, one consider the simple connected covering of X. In [79], the smoothness conditions are reduced: the class C r is replaced by the class Wqr , q > n, of functions having rth-order generalized derivatives in sense of Sobolev [131] summable with power q. A Riemannian C r -manifold is a pair (X, ds2 ) consisting of an n-dimensional manifold X and a Riemannian metric ds2 of class C r on it. An immersion z : X → E satisfying the condition I(z) = ds2 is called an isometric immersion of the Riemannian manifold (X, ds2 ) into the Euclidean space E. Two immersions z, zˆ : X → E (and the corresponding surfaces) are said to be isometric if I(z) = I(ˆ z ), and congruent or trivially isometric if they differ only by a motion of the space E, i.e., if they are related by the formula zˆ = P ·z +ω, where P = const is an orthogonal (m×m)-matrix and ω = const is a vector of E. Obviously, congruent immersions are always isometric. Generally speaking, the converse does not hold. The main problem of the theory of bendings is obtaining conditions under which isometric immersions are congruent and establishing the possibility of the existence of isometric but noncongruent immersions. One of the most general assertions of such type is the uniqueness part of the main theorem; from this standpoint, it can be formulated as follows. Theorem 9.2. Isometric C r -immersions z, zˆ : X → E, r ≥ 3, of an n-dimensional Riemannian C r−1 -manifold (X, ds2 ) into an m-dimensional Euclidean space E, 2 ≤ n < m, are congruent if and only if for any point x ∈ X, there exist orthonormal bases (νσ )pσ=1 and (ˆ νσ )pσ=1 in the normal planes Tx⊥ F ˆ σ and the torsion forms κ τ such Tx⊥ Fˆ such that the corresponding second fundamental forms IIσ and II σ and κ ˆ στ , τ, σ = 1, . . . , p with respect to these bases respectively coincide. Other general congruence criteria are presented in Sec. 9.4. 9.2. The Main Theorem in the Invariant Form. A detailed exposition of the matter discussed in this section can be found in [31, 81, 134, 135, 141, 142]. We consider a C r -immersion z : X → E, r ≥ 2, of an n-dimensional manifold X into an m-dimensional Euclidean space E. For an arbitrary point x ∈ X in the normal plane Tx⊥ F of the surface F = z(X), we fix an orthonormal basis (νσ )pσ=1 and define the symmetric bilinear mapping II : Tx (X) × Tx (X) → Tx⊥ F,

II(u, v) =

p


IIσ (u, v)νσ

∀u, v ∈ Tx (X).

σ=1

The mapping II is independent of the choice of the basis (νσ )pσ=1 and is called the vector-valued second fundamental form of the immersion z or of the surface F at the point x. In tensor notation, the vector-valued second fundamental form has the form p n

bσij dxi ⊗ dxj ⊗ νσ II = σ=1 i,j=1

and is an element of the space Tx∗ (X) ⊗ Tx∗ (X) ⊗ Tx⊥ F , where Tx∗ (X) is the cotangent bundle of the manifold X at the point x. We denote by T ∗ (X) the cotangent bundle of the manifold X and by T ⊥ F the normal bundle of the surface F ; then the form II can be considered as a C r−2 -smooth section of the bundle T ∗ (X) ⊗ T ∗ (X) ⊗ T ⊥ F . The definition of the second fundamental form can be based on the following characteristic property of it (see [73, 81]). For any vector fields U and V on the space E, we denote by ∇U V the covariant derivative of the field V in direction of the field U with respect to the connection on E determined by the 864

Euclidean metric. For vector fields u and v on X, we denote by ∇u v the covariant derivative of the field v in direction of the field u with respect to the connection induced by the metric I(z) on X. Let T (X) be the tangent bundle of the manifold X and T z : T (X) → T (E) be the tangent mapping for the immersion z (it is often called the derivative mapping or the differential and is denoted by z∗ ). Then the following relation holds: (9.8) ∇T z(u) T z(v) = ∇u v + II(u, v). We define the covariant differentiation ∇⊥ in the normal bundle of the surface F as follows. For any normal field ν of the surface F in E and any vector field u on X, we take as ∇⊥ u ν at any point x ∈ X the  ⊥ component of the field ∇T z(u) ν lying in the normal plane Tx F . Then ∇T z(u) ν = ∇⊥ u ν + A(u, ν),

(9.9)

where A : T (X) × T ⊥ (F ) → T (X) is the bilinear mapping related with the second fundamental form II by the formula I(A(u, ν), v) = −II(u, v), ν , where I(A, v) is the inner product with respect to the metric I(z) and ·, · is the inner product in E. Relation (9.8) is called the Gauss formula and relation (9.9) is called the Weingarten formula. For r ≥ 3 and vector fields u and v on the manifold X, we define the operators R(u, v) : T (X) → T (X) and R⊥ (u, v) : T ⊥ (F ) → T ⊥ (F ) as follows: R(u, v)w = ∇u (∇v w) − ∇v (∇u w) − ∇[u,v] w, ⊥ ⊥ ⊥ ⊥ R⊥ (u, v)ν = ∇⊥ u (∇v ν) − ∇v (∇u ν) − ∇[u,v] ν,

where [u, v] is the Poisson bracket acting by the rule [u, v](f ) = u(v(f )) − v(u(f )) for any C 2 -function f . The operator R(u, v) is called the curvature operator of the Riemannian manifold (X, I) and if u and v are orthonormal, then the quantity I(R(u, v)v, u) is called the sectional curvature of this manifold in the two-dimensional direction (u, v). We also define the mapping ∇u A : T (X) × T ⊥ (F ) → T (X) by the formula (∇u A)(v, ν) = ∇u (A(v, ν)) − A(∇u v, ν) − A(v, ∇⊥ u ν). Since the metric of E is Euclidean, we have the following relation for any vector fields U , V , and W on the space E: ∇U (∇V W ) − ∇V (∇U W ) − ∇[U,V ] W = 0. Setting U = T z(u), V = T z(v), and W = T z(w) and separating the tangent and normal (to F ) components, we obtain two equations. If we take the normal field ν as W , we obtain two additional equations. There are three independent equations among these four equations. Using the Gauss and Weingarten formulas (9.8) and (9.9), we can write these equations in the form I(R(u, v)w, s) = II(u, s), II(v, w) − II(u, w), II(v, s) , ∇u A(v, ν) = ∇v A(u, ν), ⊥





(9.10) 

R (u, v)ν, ν = A(u, ν ), A(v, ν) − A(u, ν), A(v, ν ) , where u, v, w, and s are arbitrary vector fields on X and ν and ν  are arbitrary normal fields for the surface F . System (9.10) is the system of the Gauss, Peterson–Codazzi, and Ricci equations in the invariant form. Now let H(X) be a locally trivial, m-dimensional vector bundle with base X (everywhere, we denote bundles and subbundles by the symbols of their total spaces). Assume that in any fiber Hx (X), an inner product ·, · x : Hx (X) × Hx (X) → R is given such that the function ξ, η is a C r -function on X for all C r -smooth sections ξ, η : X → H(X). Moreover, we assume that in the bundle H(X), there is a connection ∇ satisfying the condition u(ξ, η ) = ∇u ξ, η + ξ, ∇u η for any vector field u on X and any C 1 -sections ξ and η of the bundle H(X). The pair (·, · , ∇ ) is called the metric connection in H(X). If G(X) is a subbundle of the bundle H(X), then we can define on G(X) the so-called induced metric connection (·, · , ∇), where the inner product is taken from H(X) and ∇ is the projection of ∇ defined 865

by the relation ∇u ξ, η = ∇u ξ, η for any vector field u on X and any C 1 -sections ξ and η of the bundle G(X). If f : T (X) → G(X) is a local diffeomorphism [149] of the tangent bundle of X to the subbundle G(X), then the inner product ·, · and the metric connection (·, · , ∇ ) induce the Riemannian metric on X and the metric connection (·, · , ∇) on T (X). The connection ∇ on T (X) is the Levi-Civita connection of the induced Riemannian metric if and only if it has zero torsion. Let (X, ds2 ) be a Riemannian manifold and H(X) be the vector bundle with metric connection (·, · , ∇ ). This connection is said to be concordant with the metric ds2 if there exists a local bundle diffeomorphism f : T (X) → G(X) ⊂ H(X) such that the metric on X induced by the inner product ·, · coincides with ds2 , the induced metric connection on T (X) coincides with the Levi-Civita connection of the ds2 , and for any vector fields u and v of class C 1 , the relation ∇u v − ∇u v = ∇v u − ∇v u holds on X. In the case of a bundle with a metric connection concordant with the metric ds2 , we can formally define the second fundamental form on X by the formula II(u, v) = ∇u v −∇u v; the last condition means that the second fundamental form is symmetric. For a Euclidean space E, the inner product in E obviously generates the flat metric connection on the tangent bundle T (E). For a C 2 -immersion z : X → E, the metric connection on T (E) induces metric connections on the bundle T (E)|z(X) and on the tangent and cotangent bundles of the surface F = z(X). This determines the approach to the investigation of the Gauss, Peterson–Codazzi, and Ricci equations. The existence part of the main theorem can be represented as the following two theorems [73, 141, 142]. Theorem 9.3. Let (X, ds2 ) be an n-dimensional Riemannian manifold. The following conditions (a) and (b) are equivalent: (a) there exist a p-dimensional, locally trivial vector bundle N (X) with metric connection (·, · , ∇⊥ ) and a symmetric bilinear mapping α : T (X) × T (X) → N (X) satisfying the following conditions: ds2 (R(u, v)w, s) = α(u, s), α(v, w) − α(u, w), α(v, s) , ∇u A(v, ξ) = ∇v A(u, ξ),

(9.11)

R⊥ (u, v)ξ, η = A(u, η), A(v, ξ) − A(u, ξ), A(v, η) , where ∇ is the Levi-Civita connection of the metric ds2 , A : T (X) × N (X) → T (X) is the bilinear mapping defined by the formula I(A(u, ξ), v) = −α(u, v), ξ , and R and R⊥ are defined by the formulas R(u, v)w = ∇u (∇v w) − ∇v (∇u w) − ∇[u,v] w, ⊥ ⊥ ⊥ ⊥ R⊥ (u, v)ξ = ∇⊥ u (∇v ξ) − ∇v (∇u ξ) − ∇[u,v] ξ,

where u, v, w, and s are arbitrary vector fields from T (X) and ξ and η are arbitrary vector fields from N (X); (b) there exists an (n + p)-dimensional, locally trivial vector bundle with flat metric connection compatible with the metric ds2 . Theorem 9.4. Let (X, ds2 ) be an n-dimensional, simply connected Riemannian manifold and E be an m-dimensional Euclidean space. If an m-dimensional, locally trivial vector bundle H(X) with flat metric connection compatible with the metric ds2 exists, then there exists an immersion z : X → E such that I(z) = ds2 and the bundle H(X) is isomorphic to the bundle T (E) z(X) . For the most prevalent case where H(X) is a bundle whose fibers coincide with the vector part of the space E, the last two theorems can be found in [134, Chap. 7, Sec. 18] and for the case where m = n + 1 — in [81, Chap. 7, Sec. 7]. Two m-dimensional, locally trivial vector bundles H(X) and H  (X) with metric connections (·, · , ∇ ) and (·, ·  , ∇ ), respectively, are said to be equivalent if there exists a bundle mapping Ψ : H(X) → H  (X) 866

such that Ψ(ξ), Ψ(η)  = ξ, η and Ψ(∇u ξ) = ∇u Ψ(ξ) for any vector fields ξ and η in H(X) and any vector field u on X. The main theorem in the uniqueness part can be stated as follows. 3   Theorem 9.5. If, for C -immersions z, z : X → E, the metrics I(z) and I(z ) coincide and the bundles T (E) z(X) and T (E) z
(X) with induced metric connections are equivalent, then the immersions z and z are congruent.

For simply connected manifolds, there exists a bijective correspondence between equivalence classes of bundles with flat metric connections concordant with the given metric and classes of congruent, isometric immersions. 9.3. Main Theorem in Terms of Exterior Forms. Let U be a neighborhood on a smooth, n-dimensional manifold X. A local C r -coframe with domain U on X is an ordered set (τ i )ni=1 of linear differential forms of class C r on X, which are linearly independent at any point x ∈ U . In the case where r ≥ 1, for a given local coframe (τ i )ni=1 , we can introduce the connection forms Φij by the following formula: dτ i =

n


τ j ∧ Φij ,

Φij = −Φji ,

i, j = 1, . . . , n,

(9.12)

j=1

where d is the exterior differentiation. The set of 1-forms Φ = (Φij )ni,j=1 is called the Levi-Civita connection n  of the coframe (τ i )ni=1 . The coefficients Γijk in the expansion Φij = Γijk τ k are called the Chistoffel k=1

symbols of the connection Φ. Using the Christoffel symbols, we can construct covariant differentiation similarly to what was done in classical tensor analysis. We consider a C r -immersion z : X → E, r ≥ 1, of a manifold X into an m-dimensional Euclidean space E. In a neighborhood of any point in the surface F = z(X), there exists a local orthonormal C r−1 -frame (e1 , . . . , en , ν1 , . . . , νp ) in E such that the fields ei are tangent to F . This frame determines the local coframe (τ i )ni=1 on X by the formula dz =

n


τ i ei .

(9.13)

i=1

In this case, for the metric I(z), the following relation holds: I(z) =

n


τ i ⊗ τ i.

(9.14)

i=1

In the case where r ≥ 2, the decompositions of the differentials dei and dνσ with respect to the frame (e1 , . . . , en , ν1 , . . . , νp ) yield the Gauss and Weingarten formulas dei =

n
k=1

Φki ek

+

p


ωiσ νσ ,

dνσ = −

σ=1

n
i=1

ωiσ ei

+

p


κσρ νρ ,

(9.15)

ρ=1

where ωiσ and κσρ are 1-forms of class C r−2 called the immersion and torsion forms, respectively, i, j = 1, . . . , n, σ, ρ = 1, . . . , p. The exterior differentiation of Eq. (9.13) yields n


ωiσ ∧ τ i = 0.

(9.16)

i=1

If the decomposition of the immersion forms with respect to the forms τ i has the form ωiσ

=

n


bσij τ j ,

j=1

867

then, by the Cartan lemma (see [55, Chap. 2, Sec. 7]), Eq. (9.16) implies the symmetry relations bσij = bσji , i, j = 1, . . . , n, σ = 1, . . . , p. The second fundamental form of the surface F with respect to the normal νσ can be written in the form n
ωiσ ⊗ τ i . IIσ = i=1

For the torsion forms, the Weingarten formulas yield κσρ = −κρσ ,

ρ, σ = 1, . . . , p.

(9.17)

In the case where r ≥ 3, the exterior differentiation of the Gauss and Weingarten formulas leads to the Gauss, Peterson–Codazzi, and Ricci equations: p n

σ σ i ωi ∧ ωj = dΦj + Φik ∧ Φkj , σ=1

dωiσ = dκστ =

n
k=1 n


Φki ∧ ωkσ + ωiτ ∧ ωiσ +

i=1

i, j = 1, . . . , n,

k=1 p
τ =1 p


ωiτ ∧ κτσ ,

(9.18)

κσρ ∧ κρτ ,

ρ=1

τ, σ = 1, . . . , p.

These equations also hold for immersions of class C 2 (see [94]) if we interpret d as the generalized exterior differentiation in the sense of de Rham [46]. The main theorem of the theory of surfaces can be stated as follows. Theorem 9.6. Let X be a simply connected, smooth, n-dimensional manifold. Let (τ i )ni=1 be a local C r−1 -coframe on X, r ≥ 3, and Φij , ωiσ , and κσρ , i, j = 1, . . . , n, ρ, σ = 1, . . . , p, be linear C r−2 -forms satisfying conditions (9.12), (9.16), (9.17), and (9.18). Then there exists a C r -immersion z : X → E for which the induced metric has the form (9.14), the forms Φij are the connection forms, the forms ωiσ are the immersion forms, and the forms κσρ are the torsion forms. This immersion is unique up to transformations of the form (9.7). In this formulation, for manifolds of arbitrary topological structure, Theorem 9.6 guarantees the existence of an immersion z in any simply connected neighborhood lying in the domain of the local coframe (τ i )ni=1 . This implies the global uniqueness of this immersion. The domain of the coframe (τ i )ni=1 can be ignored as follows. Assume that a Riemannian metric ds2 on X is given. By the Lagrange method, n  τ i ⊗ τ i , where (τ i )ni=1 is in a neighborhood of any point on X, it can be reduced to the form ds2 = i=1

a local coframe having the same smoothness as the metric. The connection Φ of this coframe is called the connection of the metric ds2 . An equivalent statement of Theorem 9.6 is as follows. Theorem 9.7. Let X be an n-dimensional, smooth manifold and ds2 be a Riemannian metric of class C r−1 , r ≥ 3, on X with connection Φ = (Φij )ni,j=1 . For any C r−2 -solution {ωiσ , κσρ }, i = 1, . . . , n, σ, ρ = 1, . . . , p, of system (9.18), there exists a C r -immersion z : X → E, dim E = n + p, inducing the metric ds2 , for which the forms ωiσ are the immersion forms and the forms κσρ are the torsion forms. The immersion z is defined uniquely up to a transform of the form (9.7). The proof of this theorem in local formulation for two-dimensional manifolds and the three-dimensional Euclidean space is given in [55, Chap. 3, Sec. 5]. It can be easily generalized to the case of arbitrary dimensions. In the global formulations, this theorem was proved by Borovski [14–17] for simply connected manifolds X under minimal smoothness assumptions by using the technique of generalized exterior differentiation. Borovski proved that if a solution of system (9.18) belongs to the class Lp , where p ≥ n for n > 2 and p > n for n = 2, then the immersion z belongs to the class W21 . 868

9.4. Some General Criteria of the Congruence and Unique Definiteness. The main theorem of the theory of surfaces in all versions presented above can be considered, in some sense, as a congruence criterion of surfaces. Roughly speaking, this criterion consists of the following: two surfaces are congruent if and only if their first and second fundamental forms respectively coincide. This assertion is sufficiently precise for hypersurfaces; however, in the case where the codimension p > 1, it must be stated more carefully. The fact is that, by the main theorem, a unique (up to location in the space) surface corresponds to any solution of the Gauss, Peterson–Codazzi, and Ricci equations (certainly, if the smoothness conditions hold). However, different solutions can generate the same surfaces. For example, the second fundamental form can be changed by transformations of the basis of the normal plane such that the surface itself does not vary; therefore, the given surface can be defined by different solutions of the Gauss, Peterson–Codazzi, and Ricci equations. If we use the main theorem in the invariant form (Theorem 9.5), these difficulties are absent, but it is fairly hard to verify the conditions of the theorem and, therefore, it is not convenient to use this theorem as a congeruence criterion. The problem of describing of the class of solutions of the Gauss, Peterson–Codazzi, and Ricci equations (9.2) and (9.18) that generate congruent surfaces arises. In this connection, we indicate the paper of Nomizu [110], in which, in particular, the following assertion was proved: two immersions are congruent if and only if there exists an isomorphism of the normal bundles of the corresponding surfaces preserving the first and second fundamental forms and the second fundamental forms are parallel. Another congruence criterion is the following theorem. Theorem 9.8 (see [97]). Two C 2 -immersions z : X → E and zˆ : X → E are congruent if and only if the following conditions hold : (a) I(z) = I(ˆ z ); ˆ iσ and the torsion forms (b) in some neighborhood of any point of X, the immersion forms ωiσ and ω ρ ρ ˆ σ of the immersions z and zˆ, respectively, satisfy the relations κσ and κ ω ˆ iσ

=

p


ϕρσ ωiρ ,

κ ˆ σρ

ρ=1

p p
1
χ χ χ χ = (dϕσ ϕρ − dϕρ ϕσ ) + ϕχσ ϕερ κχε , 2 χ=1



i = 1, . . . , n, σ, ρ = 1, . . . , p,

χ,ε=1

 ϕ11 · · · ϕ1p where ϕρσ are C 1 -functions such that the matrix . . . . . . . . . . . . is orthogonal. ϕp1 · · · ϕpp In the classical notation, a similar theorem is stated as follows. Theorem 9.9. Two C 2 -immersions z : X → E and zˆ : X → E are congruent if and only if the following conditions hold : (a) I(z) = I(ˆ z ); (b) in some neighborhood of any point of X, the coefficients of the second fundamental forms bσij and ˆbσij and the coefficients of the torsion forms µρσi and µ ˆρσi of the immersions z and zˆ, respectively, satisfy the relations ˆbσ = ij

p
ρ=1

ϕρσ bρij ,

µ ˆρσi

p p
1
χ χ χ χ = (ϕσ,i ϕρ − ϕρ,i ϕσ ) + ϕχσ ϕερ µεχi , 2 χ=1

i, j = 1, . . . , n, σ, ρ = 1, . . . , p,

χ,ε=1

where ϕρσ are defined in Theorem 9.8. In the case where p = 1, Theorems 9.8 and 9.9 coincide with the main theorem in the part of uniqueness. Corollary 9.1. Two hypersurfaces of class C 2 are congruent if and only if their first fundamental forms coincide and the second fundamental forms coincide up to sign. 869

An immersion z : X → E (and the corresponding surface F = z(X)) is said to be globally rigid (or uniquely determined 1 ) by its metric in a given class of immersions (surfaces) if any immersion zˆ : X → E of this class such that I(ˆ z ) = I(z) is congruent to z. In problems of global rigidity, the most preferable class is the class of C r -immersions X → E denoted by Dr (X, E), r = 1, . . . , ∞, ω (by Dω (X, E), we denote the set of analytic immersions). The notion of continuous rigidity is a special case of the general notion of global rigidity. The set r D (X, E) is an open set (and, therefore, a submanifold; see [18, 85]) of the linear space C r (X, E) of all C r -mappings X → E. A bending of an immersion z ∈ Dr (X, E) (or of the corresponding surface F = z(X)) in the class C r is a continuous family of mappings f : (−ε, ε) → Dr (X, E), ε > 0, satisfying the following conditions: (1) f (0) = z; (2) the metric I(f (t)) is independent of t ∈ (−ε, ε). A bending of an immersion z is said to be trivial if it has the form f (t) = P (t) · z + ω(t), where for any t ∈ (−ε, ε), P (t) is an orthogonal (m × m)-matrix constant on X and ω(t) is a vector of E constant on X. An immersion z (surface F ) is said to be bendable in a given class if it admits a nontrivial bending in this class; it is said to be unbendable (or continuously rigid ) if all of its bendings in this class are trivial. As a criterion of global rigidity of immersions in the class C r , r ≥ 1, the following theorem can serve. Its proof is based on the study of Schl¨ afly equations (9.1) for z. Theorem 9.10 (see [99]). Let z : X → E be a C r -immersion, r ≥ 1, of an n-dimensional manifold X n  gij dxi ⊗ dxj on X, and into an m-dimensional Euclidean space E, which generates the metric I(z) = i,j=1

(g ij )ni,j=1 be the matrix inverse to the matrix (gij )ni,j=1 . The immersion z is globally rigid in the class C r if and only if there exist functions vσα , α = 1, . . . , m, σ = 1, . . . , p = m − n, of class C r−1 , for which any solution (uα )m α=1 of the system n
i,j=1

where

δ αβ

∂uα ∂uβ
α β + vσ vσ = δ αβ , ∂xi ∂xj p

g ij

σ=1

is the Kronecker symbol, α, β = 1, . . . , m, has the form uα =

m


pαβ z β + ω α ,

β=1

(pαβ )m α,β=1

where is an orthogonal matrix with elements constant on X, z α are the components of the immersion z with respect to a fixed, orthogonal, Cartesian coordinate system in E, and ω α = const. Almost all assertions of this section can be easily modified to the case where E is a flat space, i.e., an affine space with a nondegenerate inner product, or a space of constant curvature (hypersphere of a flat space). 10. Isometries and Bendings 10.1. Global and Continuous Rigidity of Hypersurfaces. Apparently, the first result on the global rigidity of multidimensional surfaces belongs to Beez [5, 6]. He proved that for n ≥ 3 (except for some degenerate cases), any n-dimensional surface of class C 3 in the (n + 1)-dimensional Euclidean space is uniquely determined by its metric in the class C 3 . This phenomenon was studied in papers of Killing [77], Bianchi [8, 9], Cartan [20], Thomas [148], Eisenhart [54], Dolbeaut-Lemoin [50], and others. The main result can be stated as follows. 1

In English mathematical literature, there is no common term meaning the rigidity of a surface in both senses “continuous” and “discrete.”

870

Theorem 10.1. Let z : X → E be a C 2 -immersion of an n-dimensional manifold X into an (n + 1)-din  bij dxi ⊗ dxj be the second fundamental form. If rank II(z) ≡ mensional flat space E and II(z) = i,j=1

rank(bij ) ≥ 3 at any point of X, then the immersion z is uniquely determined by its metric in the class C 2 . Note that the Beez theorem asserts the local global rigidity, i.e., the unique definiteness of an arbitrarily small neighborhood of any point, where rank II(z) ≥ 3, among all other isometric surfaces. A simple proof of this theorem can be found, e.g., in [3, 54, 135]. It is based on the following result of Thomas: if rank II(z) ≥ 3, then the system of Gauss equations (9.3a) has no more than one solution with respect to (bij ). This fact has important consequences. Since system (9.3a) is algebraic, the geometry of considered surfaces loses its differential nature and becomes one of the branches of algebraic geometry (see [25]). Smoothness conditions for the hypersurface in the Beez theorem can be weakened. For example, in [16], this theorem is proved for the hypersurfaces of class W11 (see p. 864) under the following additional condition: the forms τ i of the local coframe and the immersion forms ωi are square integrable (i.e., belong to the class L2 ). This result is especially interesting in connection with the papers of Kuiper [83, 84] and Bleecker [11]. In these papers, the following assertion is proved by using Nash’s results (see [108]): under sufficiently general conditions of a topological nature, any C 1 -surface in a Euclidean space is not globally rigid in the class C 1 . The Beez theorem implies that only hypersurfaces such that the rank of the second fundamental form does not exceed two cannot be uniquely defined (for example, in the class C 2 ) at a neighborhood of any point. Therefore, the condition rank II(z) ≤ 2 can serve as the necessary condition of bendability. Using this fact, Sbrana [122] and Cartan [20] studied the structure of bendable hypersurfaces and gave a complete classification of them. The structure of hypersurfaces admitting bendings preserving the mean curvature was studied in [101, 133]. In particular, it is proved in [101] that if an immersion z : X → E is analytic, then the corresponding surface can be one of the following: (a) a minimal hypersurface; (b) an open domain on the cylinder M 2 × Rn−2 , where M 2 is a locally bendable (with preserving mean curvature), two-dimensional surface in the three-dimensional Euclidean space; (b) an open domain on the cylinder CN × Rn−3 , where N is a locally bendable (with preserving mean curvature), two-dimensional surface of a three-dimensional sphere and CN is a three-dimensional cone over N in the four-dimensional Euclidean space. Without the analyticity condition, combinations of these three cases are possible. Bendings of hypersurfaces satisfying rank II(z) ≤ 2 were studied by Dajczer [30]. The global structure of locally bendable hypersurfaces with constant scalar curvature k = 0 was studied in [63, 64]. It was proved that any complete surface of such type is isometric to the product of the two-dimensional sphere of radius 1/k by the Euclidean plane of dimension n − 2. By Theorem 10.1, many assertions on the global rigidity and continuous rigidity of two-dimensional surfaces in the three-dimensional Euclidean space in the case of hypersurfaces of higher dimension become trivial. However, there are results with weakened conditions for the rank of the second fundamental form; they are not consequences of Theorem 10.1. The most famous among them was obtained by Sacksteder. Theorem 10.2 (see [118, 119]). Let z : X → E be a C 2 -immersion of an n-dimensional manifold X into an (n + 1)-dimensional Euclidean space E satisfying the following conditions: (1) the metric space (X, I(z)) is complete; (2) there exists a point on X at which r = rank II(z) ≥ 3; (3) the set of all points of X where II(z) = 0 is connected ; (4) there are no submanifolds in X which are homeomorphically mapped by the immersion z on a complete (n − r + 1)-dimensional plane in E. Then the immersion z is uniquely defined by its metric in the class C 2 . 871

This theorem is a consequence of another theorem of Sacksteder, which strengthens the result of Thomas mentioned on p. 871. Theorem 10.3. Let z : X → E and zˆ : X → E be two C 2 -immersions of an n-dimensional manifold X into an (n + 1)-dimensional Euclidean space E satisfying the following conditions: (1) the metric space (X, I(z)) is complete; (2) I(z) = I(ˆ z ); (3) r ≡ max rank II(z) ≥ 3; X

(4) there are no submanifolds in X which are homeomorphically mapped by the immersion z on a complete (n − r + 1)-dimensional plane in E. Then II(ˆ z ) = ±II(z) on X. In its order, this theorem is a consequence of the following sophisticated result of Sacksteder. Theorem 10.4. Let z : X → E and zˆ : X → E be two C 2 -immersions of an n-dimensional manifold X into an (n + 1)-dimensional Euclidean space E and Y be an n-dimensional submanifold in X such that any Cauchy sequence in the space (Y, I(z)) has a limit in X. Assume that the following conditions hold : (1) I(z) = I(ˆ z ); (2) r ≡ max rank II(z) < n; Y

(3) there are no submanifolds in X which are homeomorphically mapped by the immersion z on a complete (n − r)-dimensional plane in E. If II(ˆ z ) = ±II(z) on the boundary of the manifold Y , then II(ˆ z ) = ±II(z) on the whole Y . If we take a submanifold Y such that Y = X, then we obtain an analogue of the following well-known theorem: a two-dimensional developing surface in the three-dimensional Euclidean space is determined by its boundary band. The results of Sacksteder imply that any closed, convex hypersurface of class C 2 in a Euclidean space is uniquely defined by its metric in the class of C 2 -immersions. A similar result was obtained by do Carmo and Warner [49] for C ∞ -immersions into spaces of constant positive curvature. Sen’kin [125] proved the unique definiteness of general (i.e., continuous without any additional smoothness requirements) closed, convex hypersurfaces in Euclidean spaces in the class of convex surfaces and the local unique definiteness in neighborhoods of point where surfaces are strongly convex. Gorzii [60] generalized these results of Sen’kin to the case of elliptic space. For immersions into spaces of arbitrary nonzero, constant curvature, Matsuyama proved the following theorem. Theorem 10.5 (see [102]). Let z : X → V and zˆ : X → V be two C 3 -immersions with nonzero, constant mean curvature of an n-dimensional, n ≥ 3, manifold X into an (n + 1)-dimensional space V of nonzero constant curvature. If I(z) = I(ˆ z ), then zˆ and z are congruent. Dajczer and Gromol [39] proved that if a hypersurface F ⊂ E contains no open set Y × Rn−3 with unbounded Y , then the unique definiteness can be violated only along ruled bands (components). If F is a nowhere completely ruled surface (i.e., contains no domains consisting of complete straight lines) and the set of umbilic points does not divide it, then F is uniquely determined by its metric. Dajczer and Tenenblat [42] proved the unique definiteness in the class C 3 of a complete Weingarten hypersurface (i.e., a surface whose mean and scalar curvatures satisfy a functional relation) that contains no open subset of the form Y × Rn−3 , where Y is unbounded, in an (n + 1)-dimensional, n ≥ 4, Euclidean space. 10.2. Global and Continuous Rigidity of Surfaces of Codimension p > 1. For a long time, the problem on the unique definiteness of surfaces of codimension p > 1 was solving by generalizing corresponding results for hypersurfaces. The first such result, the Beez theorem (Theorem 10.1), was generalized by Allendoerfer in 1939. The central role in the Allendoerfer theorem is played by the notion 872

of the so-called type number of an immersion, which generalizes the notion of the rank of the second fundamental form to the case of codimension p > 1. The definition of the Allendoerfer type number was revised by many authors (see [26, 27, 31, 57, 81, 82, 153–163]). We formulate the definition of this notion following [81]. Let z : X → E be a C 2 -immersion of an n-dimensional manifold X in an m-dimensional Euclidean space E and F = z(X) be the corresponding surface. At an arbitrary point x ∈ X, we fix an orthonormal (with respect to the metric I(z)) basis ξ = (ξi )ni=1 of the tangent space Tx (X). Then the vectors ei = ξi (z) form an orthonormal basis e = (ei )ni=1 of the tangent plane Tx F of the surface F . We also fix an orthonomal basis ν = (νσ )pσ=1 in the normal plane T ⊥ F and denote the corresponding immersion forms (see p. 867) by ωiσ and the second fundamental form with respect to the normal νσ by IIσ . In the case where, among the forms II1 , . . . , IIp , there are exactly q linearly independent forms, q < p, the basis ν can be chosen such that IIq+1 = · · · = IIp = 0. In what follows, we assume that the basis ν is chosen precisely in this way. For this basis, we have ωiq+1 = · · · = ωip = 0, i = 1, . . . , n. The q-dimensional subspace Nx of the space Tx⊥ F generated by the vectors ν1 , . . . , νq is called the principal normal space of the immersion z (surface F ) [112]. For any fixed σ = 1, . . . , q, we define the linear mapping ω σ : Tx (X) → Tx (X) by setting σ

ω (u) =

n


ωiσ , u ξi

i=1

for any vector u =

n  i=1

ui ξi ∈ Tx (X), where ·, · is a pairing. The type number of the immersion z

(surface F ) at a point x ∈ X is the maximal integer t satisfying the following condition: in Tx (X), there exist t vectors u1 , . . . , ut such that qt vectors ω σ (uλ ), σ = 1, . . . , q, λ = 1, . . . , t, are linearly independent. One method of calculating the type number is based on Theorem 10.6 below. Recall some notions of the theory of spatial matrices [132]. Let τ = (τ i )ni=1 be the basis of the cotangent space Tx∗ (X) dual to the basis ξ and let the second fundamental forms have the form σ

II =

n


bσij τ i ⊗ τ j .

i,j=1

We consider the three-dimensional matrix B = (bσij ) of size n × n × q. The two-dimensional rank of the matrix B with respect to the left subscript is the rank of the two-dimensional (n × nq)-matrix B  obtained from B by “stretching in line” of sections i = const. Thus, the ith row of the matrix B  consists of elements bσij with the same pairs (j, σ). A submatrix of size n × t × q (t < q) of the matrix B is a three-dimensional matrix obtained from B by removal of n − t sections j = const. Theorem 10.6. The type number of an immersion z at a point x ∈ X is the maximal integer t such that there exists an (n × t × q)-submatrix of the three-dimensional matrix B = (bσij ) whose two-dimensional rank with respect to the left subscript equals qt. The Allendoerfer theorem on the unique definiteness can be formulated as follows [2, 82, 135]. Theorem 10.7. Let X be a simply connected manifold and z : X → E be a C 2 -immersion. Assume that at all points x ∈ X, the principal normal spaces have the same dimension q ≤ p and the type numbers t(x) ≥ 3 of the immersion are also the same. Then the surface F is contained in some (n+q)-dimensional subspace E n+q and is uniquely determined by its metrics in the class of C 2 -smooth immersions into the space E n+q . The assumption on the simple connectedness of the manifold X in this theorem is needed only for the assertion that F is contained in a subspace E n+q . Namely, the following theorem holds.

873

Theorem 10.8. If, at any point x ∈ X, the dimension of the principal normal space of a C 2 -immersion z : X → E coincides with the codimension p and the type number satisfies the inequality t(x) ≥ 3, then the surface F = z(X) is uniquely determined by its metrics in the class C 2 . In contrast to Theorem 10.7, Theorem 10.8 is local in the following sense: any neighborhood of any point of the surface is uniquely determined. Furthermore, with respect to the belonging to the space E n+q , the restriction for the type number in Theorem 10.7 can be weakened. The following theorem holds. Theorem 10.9. Let X be a simply connected manifold, an immersion z : X → E be such that at any point x ∈ X, the dimension of the principal normal space is q ≤ p and the type number satisfies the inequality t(x) ≥ 2. Then the surface F is contained in some space E n+q . The proof of Theorem 10.9 can be found in [2, 27] and in [135, Chap. 12, p. 362]. The proof of Theorem 10.8 can be found in [2, 162, 163]. The proof of Theorem 10.8 is contained in [81], but Lemma 2 underlying this proof must be revised. Note that Theorems 10.7–10.9 in specified papers are proved for immersions of class C 3 but they can be generalized for the case C 2 by approximation. Papers of Yanenko [153–163] occupy a particular place in the series of works devoted to the Allendoerfer theorem. The Allendoerfer theorem implies that surfaces that are not uniquely determined by their metrics in E are surfaces with type number t ≤ 2. Therefore, the inequality t ≤ 2 can serve as a necessary condition of the bendability. Yanenko obtained new necessary conditions for the bendability, studied the structure of bendable surfaces, and classified these surfaces. Classes of bendable surfaces described by Yanenko are invariant under projective transformations of the space E. The Allendoerfer theorem is the first but not unique result on the general rigidity of immersions with codimension p > 1. In many cases, the condition t(z) ≥ 3 is too restrictive for the unique definiteness. Indeed, the type number satisfies the inequality qt(z) ≤ n, where n is the dimension of the surface and q is the dimension of the principal normal space. This means, for example, that for q ≥ 2, the unique definiteness can be stated only for surfaces of dimension n ≥ 6. However, there exist simple examples of uniquely determined, four-dimensional surfaces in the six-dimensional Euclidean space, for instance, the Riemannian product of two two-dimensional spheres (see Theorem 10.13 below). In the following theorem, Allendoerfer’s restrictions on the dimension and codimension in theorems on the unique definiteness are weakened. Theorem 10.10 (see [7]). Let z : X → E be a generic C ∞ -immersion of an n-dimensional manifold X in the (n + p)-dimensional Euclidean space E. Assume that one of the following conditions holds: p ≤ n, p ≤ 3, p ≤ 4, p ≤ 6,

n ≥ 8, n = 4, n = 5, 6, n = 7.

or or or

Then the immersion z is uniquely determined in E. The word “generic” in this theorem means that coefficients of the second fundamental forms do not satisfy some algebraic relations. Do Carmo and Dajczer [47, 48] introduced another characteristic, allowing one to obtain conditions of the local unique definiteness. Let z : X → E be a C 2 -immersion of an n-dimensional manifold X ⊥ in the m-dimensional Euclidean space E and II : Tx (X) × Tx (X) → Tx F be the vector-valued second fundamental form at a point x ∈ X (we use the notation of Sec. 9.2). Let U s be an s-dimensional ⊥ ⊥ subspace, 1 ≤ s ≤ p = m − n, of the normal space Tx F and π : Tx F → U s be the orthogonal projection. We set IIU s = π ◦ II and νs (x) = max dim N (IIU s ), ⊥

U s ⊂Tx F

874

where N (IIU s ) is the null-space (see, e.g., [114]) of the mapping IIU s . The number νs (x) is called the s-zero of the immersion z at the point x ∈ X. Theorem 10.11 (see [48]). If p = m − n ≤ 5, n > 2p, and for any point x ∈ X and any integer s, 1 ≤ s ≤ p, the s-zero νs (x) satisfies the inequality νs (x) ≤ n − (2s + 1), then the immersion z is uniquely determined in the class of C 2 -immersions. For p = 1, this theorem coincides with the Beez theorem (Theorem 10.1). For 1 < p ≤ 5 and n > 2p, it is an improvement of the Allendoerfer result (Theorem 10.7). A global analogue of this theorem was obtained in [47]. Theorem 10.12. Let X be compact, n ≥ 2p, p ≤ 5, and B ⊂ X be the set of all planar points of an immersion z. Assume that the set B does not split X. If p > 1, we assume that for any point x ∈ X − B and any integer s, 1 ≤ s ≤ p − 1, the s-zero νs satisfies the condition νs (x) ≤ n − (2s + 1). Then the C 2 -immersion z : X → E is uniquely determined by its metrics in the class C 2 . The Moore theorem (see [104]) on the unique definiteness of Riemannian products of hypersurfaces is another global theorem. Let (X1 , ds21 ), . . . , (Xp , ds2p ) be Riemannian manifolds of dimension n1 , . . . , np , respectively. The Riemannian product of these manifolds is the Riemannian manifold (X, ds2 ) of dimension Np = n1 + · · ·+ np , where X = X1 × · · · ×Xp is the product of the manifolds and ds2 = ds21 + · · · + ds2p ; the summation of forms is performed in the bundle of bilinear forms over X. Let E1 , . . . , Ep be Euclidean spaces of dimension n1 + 1, . . . , np + 1. Denote by E their product, which is an (Np + p)-dimensional Euclidean space. If zσ : Xσ → Eσ , σ = 1, . . . , p, are isometric immersions, then there exists an immersion z : X → E, which induces the metric ds2 ; it is defined by the relation z(x) = z1 (x1 ) + · · · + zp (xp ) for any point x = (x1 , . . . , xp ) ∈ X, where xσ ∈ Xσ , the summation is performed in E. The corresponding surface F = z(X) ⊂ E is called the Riemannian product of the surfaces F1 = z1 (X1 ), . . ., Fp = zp (Xp ). The Moore theorem is stated as follows. Theorem 10.13. Let any surface Fσ belong to the class C 3 and be uniquely determined by its metric in the space Eσ . Then the surface F is uniquely determined by its metric in the space E. A series of fundamental assertions on the global rigidity can be found in [144] (for the proofs, see [145]). In these papers, the class of elliptic immersions was introduced. An immersion z : X → E is said to be ⊥ elliptic if for any normal vector ν ∈ Tx F , the bilinear form II(ν) = II, ν is nonzero and has at least two eigenvalues of the same sign. The ellipticity condition is used together with another nondegenerateness-type condition, which is called the Tanaka condition C. This condition consists of the following: at any point of the manifold X, the linear span of the first and second partial derivatives of the immersion z with respect to local coordinates coincides with the space E. Denote by I(z) the metric on X 2 T ∗ (X)) of the set of all induced by the immersion z; then we have the mapping I : D1 (X, E) → E(X, S+ C 1 -immersions X → E to the set of all continuous Riemannian metrices on E (and, therefore, to the set E(X, S 2 T ∗ (X)) of sections of the bundle of bilinear forms over X). Let Iz be the Frechet derivative of the mapping I : D1 (X, E) → E(X, S 2 T ∗ (X)). For an E-valued vector field U on X, we have Iz (U ) = 2 dz dU . We denote by ρ(z) the dimension of the space of solutions to the equation Iz (U ) = 0. The main result is as follows. Theorem 10.14. If X is a compact manifold, then any elliptic C 3 -immersion z : X → E satisfying possesses in the set D3 (X, E) of all C 3 -immersions X → E condition (C) and the relation ρ(z) = m(m+1) 2 a neighborhood (in the C 3 -topology) such that any two isometric immersions from this neighborhood are congruent. in this theorem means that the immersion z must possess the infinitesimal The relation ρ(z) = m(m+1) 2 rigidity of first order (see below). The main consequence of this theorem is as follows: any elliptic C 3 -immersion of a compact manifold in a Euclidean space satisfying condition (C) and possessing rigidity of first order is unbendable in the class of C 3 -immersions. It is proved in [28] that any two elliptic, isometric 875

C 3 -immersions of a compact manifold in a Euclidean space that possess C 3 -neighborhoods consisting of immersions with first-order rigidity are congruent. As a consequence of Theorem 10.14, Tanaka proved [145] the unbendability of some class of isometric immersions of a compact Hermitian space. In [76], Tanaka’s theorem was applied to the proof of the unbendability of the canonical isometric immersion of the symmetric R-space in a Euclidean space. In [75], this theorem was used for the proof of the unbendability of the classical Lie group SO(n), U(n), and Sp(n) in the n2 -dimensional space of (n × n)-matrices over the fields R and C and the body of quaternions Q, respectively. The unbendability of the groups SO(n) for n ≥ 5, U(n) for n ≥ 3, and Sp(n) for n ≥ 1 is proved. A number of papers are devoted to the problem of the unique definiteness under some additional restrictions on the varying of geometric characteristics of surfaces. Tenenblat [146] proved the local unique definiteness of three-dimensional surfaces in the six-dimensional Euclidean space in the class C ∞ under the condition that asymptotic hyperplanes of tangent spaces are preserved. Let x be an arbitrary point of a manifold X. A k-dimensional, 0 < k < n, subspace L of the tangent space Tx (X) is called the ⊥ asymptotic space for an immersion z : X → E if there exists a normal ν ∈ Tx F such that II(ν) L = 0. A k-dimensional submanifold Y of a manifold X is called an asymptotic manifold for z at a point x if the tangent space Tx (Y ) is an asymptotic subspace of the space Tx (X) (see [68–72, 146]). For an immersion z : X → E and a point x ∈ X, we denote by Cx the image of the set of all (n − 1)-dimensional subspaces under the tangent mapping Tx z : Tx (X) → Tx F . For n = 3, the set Cx is a cubic surface in the tangent plane Tx F for F considered an affine subspace of the space E. Let zˆ : X → E be an immersion isometric to z, Fˆ be the corresponding surface, Cˆp be the image of the set of (n − 1)-dimensional asymptotic subspaces for zˆ under the tangent mapping for zˆ. Then an isometry f : F → Fˆ is defined. The main result of [146] is stated as follows. Theorem 10.15. Let dim X = 3, dim E = 6, z, zˆ : X → E be isometric C ∞ -immersions, and f : F → Fˆ be the corresponding isometry. Assume that at any point x ∈ X, the sets Cx and Cˆx are nonsingular cubic surfaces in Tx F and Tx Fˆ , respectively, and Tz(x) f (Cx ) = Cˆx . Then the surfaces F and Fˆ are congruent. Ros [115] proved that if an n-dimensional surface F  with mean-curvature vector H in an m-dimensional Euclidean space is isometric to a compact surface F lying in an (n + 1)-dimensional plane and having nonnegative mean curvature in this plane, then, under the condition |H | ≤ H, the surface F  is congruent to the surface F . A similar result was obtained by Dajczer and Gromol [38] for minimal surfaces (i.e., for surfaces with zero mean-curvature vector). They proved that if (X, ds2 ) is a complete, n-dimensional, n ≥ 4, Riemannian manifold that does not contain Rn−3 as a factor and admits an isometric immersion z in the (n + 1)-dimensional Euclidean space as a minimal surface, then any isometric immersion of this manifold as a minimal surface in the m-dimensional Euclidean space, where m > n is arbitrary, is congruent to z. It was proved in [1] that two isometric hypersurfaces with the same Grassmann image can be joined by a parallel translation and a specular reflection. Borisenko [12,13] improved this result. It is proved that if an n-dimensional surface F in an m-dimensional Euclidean space has a point at which the sectional curvatures of the surface are positive, then any surface isometric to it and having the same Grassmann image differs from the initial surface by a translation and, possibly, a central symmetry. Bendings of n-dimensional surfaces with boundaries in m-dimensional Euclidean spaces under similar conditions (with conservation of a fixed section of the normal pencil) were studied by Oliker [111]. For surfaces with codimension p > 1, analogues of bendings on principal bases of two-dimensional surfaces in the three-dimensional Euclidean space (see, e.g., [74, 87, 136]) are interesting. In this connection, we note papers [116, 151], where the problem on the unique definiteness under restrictions on the varying of coordinate nets of surfaces was considered. Among papers devoted to problems on the general and continuous rigidity of surfaces of codimension ˇ p > 1 with preserved geometric characteristics, we mention Svec’s paper [139], in which the author obtained sufficient conditions under which two globally isometric, two-dimensional, smooth surfaces with 876

boundaries in the four-dimensional Euclidean space are congruent. Sufficient conditions of the unique definiteness of two-dimensional surfaces in the four-dimensional Euclidean space were also obtained by Kim [78]. The unique definiteness of two-dimensional surfaces in the five-dimensional Euclidean space was also studied in [24]. The unique definiteness of the helicoid in the class of smooth minimal surfaces on the n-dimensional Euclidean space was proved in [23]. All these results were obtained for sufficiently smooth surfaces (as a rule, the C 3 -smoothness is required). Sen’kin [124] proved the following assertion without any smoothness assumptions. Let F be an (n − 1)-dimensional, general, convex surface lying in an n-dimensional plane of the m-dimensional Euclidean space, Φ be a surface isometric to F and lying in a k-dimensional plane (n ≤ k ≤ m), r(x, y) be the spatial distance between points x and y of the surface F , and ρ(x, y) be the spatial distance between the corresponding (with respect to the isometry) points of the surface Φ. If the inequality ρ(x, y) ≥ r(x, y) holds, then the surfaces F and Φ are congruent. Papers [40, 43, 44, 70] are devoted to the problem on the extension of a bending of a given surface X to a bending of a surface of higher dimension containing X. In the terminology proposed by Yanenko (see [161–163]), this problem is called the co-bending problem. In particular, it was proved in [43] that if z : X → V n+1 is an embedding of an n-dimensional manifold X in an (n+1)-dimensional, complete, simply connected space of constant curvature, z˜ : X → V n+p is an immersion of X in an (n + p)-dimensional, complete, simply connected space of the same constant curvature, and the type number t(z) of the immersion z satisfies the condition p ≤ t(z) − 2, then z˜ can be represented as the composition of z with an isometric immersion of a domain of the space V n+1 . In [44], this result was generalized to the case where V n+1 and V n+p have distinct curvatures. It was proved in [40] that if X is a compact, n-dimensional Riemannian manifold, n ≥ 5, and z, z˜ : X → E are isometric immersions of X in the (n + 2)-dimensional Euclidean space, then there exists an open, everywhere dense set on X such that on any connected component of it, either z and z˜ are congruent or they can be extended to noncongruent isometric mappings Z, Z˜ : Y → E, where either Y is an (n + 1)-dimensional plane and Z and Z˜ are regular or Z and Z˜ are hypersurfaces described by Sbrana [122] and Cartan [20]. Conformally flat submanifolds of Euclidean spaces were considered from the standpoint of bendability in [32, 33]. It was proved that in the case where the dimension n ≥ 5 and the codimension p = 2, conformally flat submanifolds can be divided into three classes. One of these classes consists of locally unbendable surfaces; surfaces of this class are explicitly described. It is proved in [152] that any isometric immersion of an n-dimensional sphere in the (n + 2)-dimensional Euclidean space can be bent to the standard embedding of this sphere in the (n + 1)-dimensional Euclidean space.

11. General Problems of the Theory of Infinitesimal and Analytic Bendings 11.1. Definitions of Infinitesimal and Analytic Bendings. The first definitions of infinitesimal bendings of two-dimensional surfaces of the three-dimensional Euclidean space are contained in papers published in the 19th century (see [103]). In these papers, as a rule, infinitesimal bendings and bendings were not distinguished. Darboux distinguished between these two notions at the end of the 19th century (see [45]). Despite the vast number of results concerning infinitesimal bendings of two-dimensional surfaces in the three-dimensional Euclidean space, the definition of the infinitesimal bending itself is still being improved (see [29, 66, 117]). In the case of multidimensional surfaces, the language of fiber bundles is convenient. The definition stated below is contained in [95]. The main idea of this definition can be expressed by the following figurative phrase of N. V. Efimov: “The theory of infinitesimal bendings is the differential of the theory of bendings.” Any fiber bundle with base X and total space P (X) we denote by the same symbol P (X). We denote by Px (X) the fiber over a point x ∈ X and by E r (X, P (X)) the set of all C r -sections of the bundle P (X), r ≥ 0. In the case where P (X) is a vector bundle, the set E r (X, P (X)) can be naturally 877

considered as a vector space and, therefore, as a C ∞ -manifold modeled by a Banach space (for the theory of infinite-dimensional manifolds, see [18, 85]). We consider a C r -path γ : (−ε, ε) → X, ε > 0, on an n-dimensional manifold X passing through a point x ∈ X, γ(0) = x. We denote by C r (x) the set of all functions belonging to the class C r in some r is neighborhood of the point x. The mapping v : C r (x) → R defined by the formula v(f ) = d fdt(γ(t)) r t=0

(r)

called the r-tangent vector for the path γ at the point x. The linear span Tx (X) of all r-tangent vectors   r (n+k−1)! of all paths passing through x is a (n−1)! k! -dimensional vector space; it is called the r-tangent k=1

(r)

space for X at the point x. The vector bundle whose fiber over the point x is the vector space Tx (X) is called the r-tangent bundle over the manifold X. In the case where r = 1, we obtain the ordinary tangent bundle T (X). Let P (X) be an arbitrary C r -bundle over X; we denote by P (r) (X) the bundle with base X whose fiber over a point x ∈ X is the total space T (r) (Px (X)) of the r-tangent bundle over the fiber Px (X). The bundle P (r) (X) is called the r-derivative bundle of the bundle P (X). Let V (X) be a vector C ∞ -bundle and θ ∈ E r (X, V (X)), r ≥ 0, be its section. A C s -deformation, s ≥ 0, of the section θ is a C s -mapping f : (−ε, ε) → E r (X, V (X)) such that f (0) = θ. If s = 0, then a deformation is said to be continuous (cf. p. 870). We say that a deformation belongs to the class C ∞ if it belongs to C s for any s = 0, 1, . . . . A fixed deformation of a section θ is denoted by {θt }t∈(−ε,ε) or {θt } and its value at a fixed point t ∈ (−ε, ε) is denoted by θt . A deformation is said to be analytic if it belongs to the class C ∞ and at any point x ∈ X it can be represented by a power series θt (x) = θ(x) + 2

∞


δ s θ(x)ts

(11.1)

s=1

converging in (−ε, ε) with coefficients δ s θ ∈ E r (X, V (s) (X)). Now let P (X) be a C r -subbundle of a vector C r -bundle V (X) and θ ∈ E r (X, P (X)). A C s -deformation, 0 ≤ s ≤ ∞, of a section θ in P (X) is a mapping f : (−ε, ε) → E r (X, P (X)), which is a C s -deformation of the section θ in E r (X, V (X)). A deformation {θt }t∈(−ε,ε) of the section θ in E r (X, P (X)) is said to be analytic if it is a C ∞ -deformation and at any point x ∈ X it can be represented by a power series of the form (11.1) converging in (−ε, ε) with coefficients δ s θ ∈ E r (X, P (s) (X)). The canonical embedding of fibers Px (X) ⊂ Vx (X) allows one to consider the value δ s θ at any point x ∈ X as a vector in Vx (X). The usage of the notion of the s-derivative bundle in the definition of an analytic deformation is stipulated by the fact that for a fixed point x ∈ X, an analytic deformation {θt } defines an analytic path (−ε, ε) → Px (X) ⊂ Vx (X) and ds θt (x) s δ θ(x) = , s = 1, 2, . . . . 2s! dts t=0 For a given deformation {θt }, the section δ s θ ∈ E r (X, P (s) (X)) defined by this formula is called the sth variation of the section θ and the operator δ s : E r (X, P (X)) → E r (X, P (s) (X)) is called the variation operator of sth order. We use the notation δ 0 θ = θ and δ 1 θ = δθ. A class of deformations defining the same variations δ 1 θ, . . . , δ l θ is called an infinitesimal deformation of lth order of the section θ (for the definition of infinitesimal deformations, see also [164, 165]). An infinitesimal deformation of lth order is uniquely defined by a deformation of the form θt (x) = θ(x) + 2

l


δ s θ(x)ts + o(l),

s=1

→ 0 as t → 0. Therefore, infinitesimal deformations of lth order can where o(l) ∈ E r (X, V (X)) and o(l) tl be considered as l-jets of analytic deformations. The definition of infinitesimal deformations in terms of jets was formulated by Klimentov [80]. In what follows, discussing infinitesimal deformations of lth order, 878

we do not exclude the case where l = ∞; by an infinitesimal deformation of infinite order, we mean an analytic deformation. Identifying a mapping h : X → E with the section x → (x, h(x)) of the trivial bundle X × E, we can extend above definitions to the case of mappings of class C r (X, E), in particular, immersions. A deformation of an immersion z : X → E generates a deformation of any tensor field θ on X expressed by z; therefore, any infinitesimal deformation of the immersion z generates an infinitesimal deformation of the tensor field θ. In particular, a deformation {zt } generates a deformation {I(zt )} of the metric I(z). An infinitesimal deformation of lth order (or analytic deformation if l = ∞) of an immersion z is called an infinitesimal bending of lth order (respectively, an analytic bending) if, for the infinitesimal deformation of the metric I(z) generated by it, all variations up to order l vanish. Often, instead of an infinitesimal bending of an immersion, one speaks about the infinitesimal bending of the corresponding surface. A variation δ s z : X → E is called a bending field of sth order of a surface F . 11.2. Infinitesimal Motions and Trivial Infinitesimal Bendings. Since I(z) = dz 2 (on the right-hand side, the scalar square in E is meant), the condition for variations of the metric under infinitesimal bending of the surface yields the following system for bending fields: s
dδ α z · dδ s−α z = 0, s = 1, . . . , l. (11.2) α=1

For l < ∞, any solution {δ s z}ls=1 of this system defines an infinitesimal bending of lth order of the immersion z. For l = ∞ (in the case of analytic bending), the convergence of the series z(x) + 2

∞


δ s z(x)ts

s=1

in (−ε, ε) at any point x ∈ X is additionally required. System (11.2) always has solutions of the form δsz =

s


Ωα · δ s−α z + ω s ,

s = 1, . . . , l,

(11.3)

α=1

 where Ωs are arbitrary, constant on X bivectors in 2 E, ω s are arbitrary, constant on X vectors in E, and the dot means the inner product of a bivector and a vector [22]. Fields of this form describe infinitesimal motions of the surface F in E as a rigid body. We say that an infinitesimal bending of lth order contains an infinitesimal bending of order k − 1 > 0 if the variations δ s z of order 1, . . . , k − 1 have the form (11.3). If representation (11.3) is valid for all bending fields {δ s z}ls=1 , then the corresponding infinitesimal bending of order l is called an infinitesimal motion of order l. For an infinitesimal bending of lth order containing an infinitesimal motion of order k − 1, by adding a motion, the corresponding deformation can be reduced to the form zt = z +

l


δ s zts + o(l),

s=k

which is called an infinitesimal bending of order (k, l) (see [53]). By classical definitions (see, e.g., [16]), an infinitesimal bending of order l is said to be nontrivial if its field δ 1 z is not an infinitesimal motion of first order. In other words, a nontrivial infinitesimal bending of order l does not contain motions of any order. In classical papers, there is no explicit definition of trivial infinitesimal bending of arbitrary order l > 1; by the negation principle, one can propose the definition of a trivial infinitesimal bending of order l as an infinitesimal bending containing an infinitesimal motion of first order. However, this definition contradicts some natural connections between infinitesimal and analytic bendings (for details, see [19,53]). Note (see [53]) that an infinitesimal bending of order l is called a nontrivial infinitesimal bending of order (k, l), where k is the minimal number such that the infinitesimal 879

bending considered does not contain any motion of order k. In what follows, the classical triviality of an infinitesimal bending of order l means that it contains an infinitesimal motion of first order. We say that an immersion z : X → E (and the corresponding surface F = z(X)) possesses rigidity of lth order if any of its infinitesimal bendings of lth order is trivial (or (1, l)-trivial, in terms of [117]). In [90], the following theorem is proved. Theorem 11.1. An infinitesimal bending of finite order l of a C 1 -immersion z : X → E is an infinitesimal motion of order l if and only if the immersion z possesses rigidity of order 1. An immersion z is said to be analytically unbendable if any analytical bending of it is an analytical motion. The relationship between the analytic unbendability and the rigidity of two-dimensional surfaces in the three-dimensional Euclidean space was studied by Efimov [51, 53]; a multidimensional analogue of the main result of Efimov was obtained in [90] (see also [117]). Theorem 11.2. If an immersion z : X → E possesses rigidity of first or second order, then it is analytically unbendable. One natural approach to the definition of trivial infinitesimal bendings is the vanishing of variations of spatial distances between points on a surface. This approach was used by Pogorelov [113] for the definition of rigidity in Riemannian spaces; it is compatible with the approach used by Efimov in [53]. To distinguish between this notion and the triviality in the sense of the above definition, we will use the term “ρ-triviality.” An infinitesimal bending of lth order of a surface F = z(X) is said to be ρ-trivial if the first variation of the spatial distance in E between any two points on the surface vanishes. The question of whether the notions of trivial and ρ-trivial infinitesimal bendings coincide naturally arises. The answer is given by the following theorem, which states that these notions are distinct. Theorem 11.3 (see [92]). An arbitrary ρ-trivial infinitesimal bending of first order of an n-dimensional surface of class C 1 in an m-dimensional space E is trivial if and only if this surface is not contained in any (m − 1)-dimensional plane. Note that for higher-order infinitesimal bendings, the relationship between the vanishing of higher-order variations of spatial distances and the presence of infinitesimal motions of order >1 in infinitesimal bendings has not yet been studied. Let z : X → E be a C 2 -immersion and (νσ )pσ=1 be an orthonormal C 1 -frame in the normal bundle ⊥ T F of the corresponding surface, p = m − n. We denote by IIσ the second fundamental form of the surface F with respect to the normal νσ and by κστ the torsion form for the normal νσ with respect to the normal ντ (see p. 863). Since these forms can be expressed through z, any infinitesimal bending of the immersion z generates variations of these forms. The following theorem serves as an analogue of the congruence criterion (Theorem 9.8) in the theory of infinitesimal bendings. Theorem 11.4. An infinitesimal bending of order l of a surface F in E contains an infinitesimal motion of order h, 1 ≤ h ≤ l, if and only if the first h variations of the second fundamental forms and the torsion forms have the form p s

α ϕστ δ s−α IIτ , δ s IIσ = α=1 τ =1

 p  s−1

α s−α α τ s−α χ s τ σ χ σ K τ ψ χ − ϕχ δ κσ , δ κσ = K τ + s

α=1 χ=1 s

s

s

where ϕτσ = −ϕστ are arbitrary functions of class C 1 on X, ψ στ are functions defined by the relations s

ψ στ

=

s ϕστ

p s−1

α s−α + ϕσχ ψτχ , α=1 χ=1

880

1

1

ψ στ = ϕστ ,

s

s

and K τσ = −K στ are continuous, linear differential forms defined by the relation s

K στ

=

s dϕστ

p  
s s + ϕχτ κχσ − ϕχσ κχτ , χ=1

where s = 1, . . . , l and τ, σ = 1, . . . , p. The proof of Theorem 11.4 can be found in [98]. In the classical notation, it is contained in [93]. 11.3. Rotation Fields and the Main System of Equations. In the theory of infinitesimal bendings of two-dimensional surfaces in the three-dimensional Euclidean space, the key role is played by systems of rotation vectors. The constancy of the first vectors of this system guarantees that the corresponding infinitesimal bending is trivial [51]. In the multidimensional case, when appropriate analogues of the cross product of two vectors are absent, it is convenient to use bivectors instead of rotation vectors. This was done in [88] for first-order infinitesimal bendings of a two-dimensional surface in the four-dimensional space and in [93] for the general case. The system of rotation bivectors is introduced by the following theorem. Theorem 11.5. For a given infinitesimal bending of order l, 1 ≤ l ≤ ∞, of a C r -immersion z : X → E, ⊥ r ≥ 1, and the infinitesimal deformation of a C r−1 -frame (νσ )pσ=1 of the normal bundle T F of the surface F = z(X) generated by this infinitesimal bending, there exists a unique system of bivector fields (V s )ls=1 , V s ∈ E r−1 (X, ∧2 E), such that s

dδ z =

s


α

V dδ

α=1

s−α

z,

s

δ νσ =

s


V α δ s−α νσ ,

α=1

where σ = 1, . . . , p = m − n and s = 1, . . . , l. An element V s of the system of bivector fields defined by this theorem is called a rotation field of order s. In the theory of infinitesimal bendings of two-dimensional surfaces in the three-dimensional Euclidean space, the following fact is often used: derivatives of a rotation field of sth order with respect to local coordinates on the surface are expressed by rotation fields of order < s and variations of coefficients of the second fundamental form of order ≤ s (see, e.g., [51]). A similar result for n-dimensional surfaces in m-dimensional spaces of constant curvature was obtained in [93]; it allowed one to prove an analogue of the main theorem of the theory of surfaces in the theory of infinitesimal bendings. For an arbitrary C r -immersion z : X → E, r ≥ 2, we consider on X the system of equations p s



s−α δ α bσik δ s−α bσjh + δ α bσjh δik

α=1 σ=1

 [k,h]

= 0,

  p s

s σ α τ s−α σ s−α τ α σ δ bij,k + (δ bij δ µτ k + δ bij δ µτ k ) α=1 τ =1

= 0,

(11.4)

[j,k]

  p s 
n

s τ kh α τ s−α σ s−α τ α σ α τ s−α ρ s−α τ α ρ δ µσi,j − g (δ bki δ bjh + δ bki δ bjh ) + (δ µρi δ µσj + δ µρi δ µσj ) α=1

k,h=1

ρ=2

= 0,

[i,j]

where δ s bσij = δ s bσji and δ s µτσi = −δ s µστi are unknown functions, δ 0 bσij = bσij , δ 0 µτσi = µτσi , s = 1, . . . , l, i, j = 1, . . . , n, τ, σ = 1, . . . , p, and the symbol (∗)[i,j] denotes the alternation of the expression ∗ with respect to the subscripts i and j; we use the notation of Sec. 9.1. This system can be obtained by the formal, l-multiple variation of the Gauss, Peterson–Codazzi, and Ricci equations (9.3). Following Efimov [51], this system is called the main system of the theory of infinitesimal bendings and the following theorem is called the main theorem of this theory. 881

Theorem 11.6. If a manifold X is simply connected, r ≥ 3, and l < ∞, then to any C r−2 -solution {δ s bσij , δ s µτσi }ls=1 of system (11.4), an infinitesimal bending of order l of an immersion z and the infinitesimal deformation of a C r−1 -frame (νσ )pσ=1 of the normal bundle of the surface F correspond such that the variations of coefficients of the second fundamental forms and the torsion forms coincide with the corresponding δ s bσij and δ s µτσi . In addition, the bending field δ 1 z of first order is defined by the solution  uniquely up to a term of the form Ω1 z + ω 1 , where Ω1 is an arbitrary constant bivector in 2 E and ω 1 is an arbitrary constant vector in E. Any bending field of order s, s = 2, . . . , l, is defined by the solution and the fields δ 1 z, . . ., δ s−1 z uniquely up to terms of the form (11.3). Using exterior forms, we can rewrite the main system of the theory of infinitesimal bendings in the following form (see [95, 96]): p s



(δ α ωiσ ∧ δ s−α ωjσ + δ s−α ωiσ ∧ ωjσ ) = 0,

α=1 σ=1

dδ s ωiσ =

n


Φki ∧ δ s ωkσ +

k=1

dδ s κστ =

s 
n
α=1

δ 0 ωiσ

n


τ i ∧ δ s ωiσ = 0,

i=1 p s



(δ α ωiτ ∧ δ s−α κτσ + δ s−α ωiτ ∧ δ α κτσ ),

α=1 τ =1 δ s κστ + δ s κτσ

(δ α ωiτ ∧ δ s−α ωiσ + δ s−α ωiτ ∧ δ α ωiσ ) +

(11.5a) (11.5b) (11.5c)

= 0, p


(11.5d)  (δ α κσρ ∧ δ s−α κρτ + δ s−α κσρ ∧ δ α κρτ ) , (11.5e)

ρ=1

i=1

δ 0 κστ

= and = are the immersion and torsion forms, δ s ωiσ and δ s κστ are unknown where 1-forms, d is the generalized exterior differential, s = 1, . . . , l, i, j = 1, . . . , n, and τ, σ = 1, . . . , p. This system can be obtained by the formal variation of Eqs. (9.16)–(9.18). The main theorem of the theory of infinitesimal bendings in terms of exterior forms was proved under weaker restrictions on the smoothness of an immersion than Theorem 11.6. ωiσ

κστ

Theorem 11.7. If a manifold X is simply connected, r ≥ 2, and l < ∞, then to any C r−2 -solution {δ s ωiσ , δ s κστ }ls=1 of system (11.5), an infinitesimal bending of order l of an immersion z corresponds such that δ s ωiσ and δ s κστ are the sth variations of the immersion and torsion forms. In addition, the bending field of first order has the form  x x 
p p n
1
1 σ 1 1 σ δ ωi [ei , νσ ] + δ κτ [ντ , νσ ] dz + Ω1 · z + ω 1 , δ z(x) = 2 x0 x0

i=1 σ=1

τ,σ=1

where x0 is an arbitrary point on X and each of the integrals is taken along an arbitrary path connecting x0 with a point x ∈ X (and is independent of this path), (ei )ni=1 is a local orthonormal C r−1 -frame in the tangent bundle T F of the surface F = z(X), (νσ )pσ=1 is a local orthonormal C r−1 -frame in the normal  bundle of this surface, Ω1 is an arbitrary constant bivector in 2 E, and ω 1 is an arbitrary constant vector in E. Any bending field δ s z of order s > 1 is defined by the solution and the bending fields δ 1 z, . . ., δ s−1 z uniquely up to a term of the form (11.3). Theorems 11.7 and 11.4 imply the following analogue of Efimov’s theorem from [53]. Theorem 11.8. Let an infinitesimal bending of order l of a C 2 -immersion z : X → E contain an infinitesimal motion of order h, 1 ≤ h ≤ l, and δ s ωiσ and δ s κστ be the corresponding variations of the immersion and torsion forms, s = 1, . . . , l, i = 1, . . . , n, and τ, σ = 1, . . . , p. Then there exist an infinitesimal bending of order l of the immersion z and the infinitesimal deformations of the frames (ei )ni=1 in the tangent 882

⊥

bundle T F and (νσ )pσ=1 in the normal bundle T F such that the variations of the immersion and torsion forms coicide with δ s ωiσ and δ s κστ , respectively, and the bending fields of order up to h vanish. It was proved in [96] that if the dimension q of the principal normal space of an immersion z is constant on X and the type number of this immersion (see p. 873) satisfies the condition t(z) ≥ 3, then Eq. (11.5e) is a consequence of Eq. (11.5c). If t(z) ≥ 4, then system (11.5) is a consequence of the system consisting of Eqs. (11.5a) and (11.5b) in the following sense: for any solution {δ s ωiσ }ls=1 of system (11.5a), (11.5b), there exists a unique solution {δ s ωiσ , δ s κστ }ls=1 of system (11.5) with the same first element. Most of the results presented in this section can be generalized to the case of spaces of constant curvature. 11.4. Projective Invariance of the Infinitesimal Rigidity of First Order. By the well-known Darboux–Sauer theorem [51], the property of the infinitesimal rigidity of first order for a two-dimensional surface in the three-dimensional Euclidean space is projectively invariant. A similar result for multidimensional surfaces was obtained by Yanenko [163]. The corresponding theorem is stated as follows. Theorem 11.9. Under an arbitrary projective transformation of the m-dimensional Euclidean space, any surface of class C 1 possessing infinitesimal nonrigidity of first order is transformed to a surface also possessing infinitesimal nonrigidity of first order. Relationships between normal field, bending fields, and first-order rotation fields of a given surface and its image under a projective transformation were established in [91]. Let E be an m-dimensional Euclidean space completed by an improper hypersurface. Let us fix in E an origin and a unit vector e. To any point with position vector z, we assign the point with position vector zˆ defined by the formula z+e − e; (11.6) zˆ = z·e therefore, we obtain a transformation of the space E. Any projective transformation of the space E can be reduced to the form (11.6) by imposition of affine transformations. For an arbitrary C 1 -immersion z : X → E, we define the immersion zˆ : X → E by formula (11.6). For an arbitrary normal field ν of the surface F = z(X), an arbitrary first-order bending field δz of this surface, and the corresponding rotation field V , we set δz − e(δz(z + e)) ν − e(ν(z + e)) , δˆ z= , Vˆ = V + [V · (z + e) − δz, e] . νˆ = z·e z·e  where [∗, ∗] is the exterior product in 2 E. The fields νˆ, δˆ z , and Vˆ are the normal field, the first-order bending field, and the rotation field of the immersion zˆ generated by it, respectively. In addition, the field δˆ z is trivial if and only if the field δz is trivial. 12. Theorems on Infinitesimal Rigidity and Nonrigidity As in the theory of infinitesimal bendings of two-dimensional surfaces in the three-dimensional Euclidean space, the main problem of the theory of infinitesimal bendings of multidimensional surfaces is the problem on the rigidity (of given order) of a surface under certain conditions imposed on its geometric characteristics. Since the multidimensional case attracted the attention of geometers relatively recently, there are fewer results than in the three-dimensional case. 12.1. Theorems on Infinitesimal Rigidity and Nonrigidity of Hypersurfaces. Since the first-order rigidity of a surface implies the rigidity of higher orders, the theorems on the first-order rigidity and nonrigidity are the most interesting. It is known (see, e.g., [58, 59]), that a hyperplane possesses first-order nonrigidity. This fact implies the following theorem on the first-order nonrigidity for surfaces contained in hyperplanes. Theorem 12.1. Any surface of class C 1 lying in an (m − 1)-dimensional plane of an m-dimensional Euclidean space E possesses first-order nonrigidity in the space E. 883

The proof of this theorem is immediately implied by Theorem 11.3. In [10], Bishop considered infinitesimal second-order bendings of hyperplanes. For such infinitesimal bendings, the structure of the first-order bending field can be described. An example of simple rigidity theorems is the infinitesimal analogue of the Beez theorem (Theorem 10.1). We say that an immersion z : X → E of an n-dimensional manifold X in an m-dimensional Euclidean space E possesses local first-order rigidity if for any neighborhood Ux of any point x ∈ X, the restriction z|Ux possesses first-order rigidity. (Note that this definition differs from the definition of “infinitesimal rigidity in the small” given by Efimov [52]: the rigidity in the small means that a given point possesses an infinitesimally rigid neighborhood.) Theorem 12.2 (see [89]). If at any point of an n-dimensional surface of class C 3 in an (n+1)-dimensional flat space, there exist at least two two-dimensional directions, determined by curvature lines, such that the sectional curvature of the surface in these directions does not vanish, then this surface possesses local first-order rigidity in this space. The requirement that at any point of the surface, there exist two two-dimensional directions, determined by curvature lines, in which the sectional curvature of the surface is nonzero is equivalent to the condition rank II ≥ 3 for the rank of the second fundamental form in the Beez theorem. In [41], the following infinitesimal analogue of the Sacksteder theorem (Theorem 10.2) was proved. Theorem 12.3. Let z : X → E be a C 3 -immersion of an n-dimensional, n ≥ 3, compact manifold X in an (n + 1)-dimensional Euclidean space E. If the surface z(X) contains no flat domains of dimension n, then it possesses first-order rigidity in E. In [126], Sen’kin obtained infinitesimal analogues of results on the general rigidity from [125]. He proved that any general, convex hypersurface of an m-dimensional Euclidean space containing no flat domains of dimension m−1 possesses first-order rigidity in a neighborhood of any point which does not lie in a flat domain of dimension m−2 and m−3. If a hypersurface contains (m−1)-dimensional flat domains, then it possesses first-order rigidity in neighborhoods of the above-said points outside (m − 1)-dimensional flat domains. In [61], these results were generalized to the case of elliptic spaces. In [65], infinitesimal first-order bendings of hypersurfaces with boundaries, which can be uniquely projected on a domain of a hyperplane of the Euclidean space, were considered. For the case of gliding-type boundary conditions, rigidity theorems of such hypersurfaces were obtained. It was proved that this class of rigid surfaces contains surfaces with rank II ≤ 2. In [137] (see also [140]), necessary conditions for a first-order bending field of a hypersurface were studied, under which a global isometry of this hypersurface, generating this field, exists. In [107], the rigidity of the hypersurface in E n+2 that is the product of a strictly convex, compact ∞ C -hypersurface in E n+1 and a circle was proved. The majority of these results can be generalized to the case of spaces of arbitrary constant curvature. There are results on the rigidity of hypersurfaces in an arbitrary Riemannian space. For example, it is proved in [86] that if the Ricci curvatures of an (n + 1)-dimensional Riemannian space are negative at any point of a hypersurface of this space, then this hypersurface possesses first-order infinitesimal rigidity in this Riemannian space in the following sense: any bending field of it coincides with some Killing field of the space. In this paper, a rigidity theorem for a hypersurface with fixed boundary in a Riemannian space is proved and all bending fields of a completely geodesic hypersurface with fixed boundary are described. In [143], it is proved that if a hypersurface of a Riemannian space has parallel second fundamental form and is not minimal, then the existence of a non-normal vector of an affine, infinitesimal, first-order deformation of this surface implies the existence of a nonzero bending field on this surface. 12.2. Theorems on Infinitesimal Rigidity and Nonrigidity for Surfaces with Codimension p > 1. Results listed in the preceding section show that in the case of hypersurfaces, the nonrigidity is an exceptional phenomenon. On the other hand, it is intuitively clear that surfaces with sufficiently 884

high codimension possess first-order nonrigidity. A strict proof of this fact based on ideas of Cartan [21] and Nash [109] was performed by Jacobowitz [69]; he proved that any n-dimensional surface of class C 2 possesses first-order infinitesimal nonrigidity in an n(n + 3)/2-dimensional Euclidean space. Therefore, the most interesting theorems are theorems on rigidity and nonrigidity of surfaces whose codimension p satisfies the condition 1 < p < n(n + 1)/2. Tenenblat [147] showed that under some nonsingularity-type conditions in the case of analytic immersions, the codimension p = n(n + 1)/2 can be decreased to p = n(n − 1)/2. It is proved that if X is an analytic, n-dimensional manifold, E is an n(n + 1)/2-dimensional Euclidean space, and z : X → E is an analytic immersion, then for any point x ∈ X such that the tangent space to X at x contains a nonasymptotic (n − 1)-dimensional subspace for z (see p. 876), there exists a neighborhood U on X such that the surface z(U ) possesses first-order nonrigidity (cf. Theorem 12.2). In [89], an infinitesimal analogue of the Moore theorem (Theorem 10.13) on the general rigidity of the Riemannian product of surfaces is proved. Using the notation introduced on p. 875, we can state this result as follows. Theorem 12.4. Let Fσ , σ = 1, . . . , p, be surfaces of class C 3 possessing first-order rigidity in the flat spaces Eσ containing them, respectively. Then the Riemannian product F of these surfaces possesses first-order rigidity in the space E, which is the product of the spaces E1 , . . . , Ep . The analysis of system (11.5) for immersions with sufficiently large type number allowed Markov [98] to prove the following analogue of the Allendoerfer theorem (see Theorem 10.7 on p. 873). Theorem 12.5. If the type number of a C 2 -immersion z : X → E satisfies the inequality t(z) ≥ 3 and the dimension of the principal normal space at any point of the manifold X coincides with the codimension of the surface z(X), then the immersion z possesses local first-order rigidity. This result without the condition on the dimension of the principal normal space was obtained in [41]. However, the notion of the type number used in this paper differs from that used in Theorem 12.5. The definition of the type number used in [41] can be found in [31]. In [127], infinitesimal bendings of a (k + 1)-dimensional cone with k-dimensional directrix in an (n + 1)-dimensional Euclidean space E n+1 were considered. It was proved that if the directrix possesses first-order rigidity in the space E n ∈ E n+1 containing it and the type number of it satisfies the inequality t ≥ 2, then the cone possesses first-order rigidity in E n+1 . This and Theorem 12.5 imply that if the k-dimensional directrix of a (k + 1)-dimensional cone of class C 2 lies in a space E n ∈ E n+1 and has type number t ≥ 3, then the cone possesses first-order rigidity in the space E n+1 . In [100], infinitesimal bendings of a surface glued from (generally speaking, distinct) C 2 -surfaces F + and F − of dimensions n+ and n− , respectively, by a k-dimensional surface γ of class C 2 were studied. It was proved that if the surfaces F + and F − possess first-order rigidity in an m-dimensional space E and at any point of the surface γ, the dimension of its principal normal space coincides with the codimension, then the surface F + ∪ F − ∪ γ possesses first-order rigidity in E. In [138], infinitesimal second-order bendings of two-dimensional surfaces with boundaries in the five-dimensional Euclidean space were considered. Under sufficiently general assumptions, it was proved that the triviality of an infinitesimal bending on the boundary implies the triviality of it on the whole surface. 12.3. Infinitesimal Bendings of Fibered Surfaces. In [93, 98], a sufficiently wide class of multidimensional surfaces called fibered surfaces is described. This class contains, in particular, Riemannian products. Some rigidity and nonrigidity theorems and theorems on analytic unbendability for surfaces of this class are proved. We recall the definition of a fibered surface. Let X1 , . . . , Xq be C ∞ -manifolds of dimension n1 , . . . , nq , respectively, and E1 , . . . , Eq be Euclidean spaces of dimensions m1 , . . . , mq , respectively, 2 ≤ nλ ≤ mλ , where λ = 1, . . . , q. We denote by Yq the products Yq = X1 × · · · × Xq of the manifolds and by Sq the products Sq = E1 × · · · × Eq of the Euclidean spaces; in addition, Y1 = X1 and S1 = E1 . Obviously, Sq is 885

a Euclidean space of dimension Mq = m1 + · · · + mq . For any λ = 1, . . . , q, we set Yλ = X1 × · · · × Xλ and Sλ = E1 × · · · × Eλ . If x1 ∈ X1 , . . ., xq ∈ Xq , then we set yλ = (x1 , . . . , xλ ); therefore, yλ+1 = (yλ , xλ+1 ). We consider a C r -mapping zq : Yq → Sq , r ≥ 1. It is called a q-multiple fibered immersion if at any point yq = (x1 , . . . , xq ) ∈ Yq , the decomposition zq = ρ1 (x1 ) + ρ2 (y2 ) + · · · + ρq (yq )

(12.1)

holds, where ρ1 : X1 → E1 is a C r -immersion and for any µ = 1, . . . , q−1, the mapping ρµ+1 : Yµ+1 → Eµ+1 is a C r -mapping such that for any point yµ ∈ Yµ , the mapping ρyµ : Xµ+1 → Eµ+1 defined by the formula ρyµ (xµ+1 ) = ρµ+1 (yµ , xµ+1 ), C r -immersion.

The surface Fq = zq (Yq ) is called a fibered surface. For any λ = 1, . . . , q, we set is a zλ = ρ1 (x1 ) + ρ2 (y2 ) + · · · + ρλ (yλ ). Hence, zλ : Yλ → Sλ is a λ-multiple fibered immersion. Any surface Fλ = zλ (Yλ ) ⊂ Sλ is called a base of the surface Fλ+1 and any surface Φyλ = ρyλ (Xλ+1 ) ⊂ Eλ+1 is called a fiber of this surface over a point yλ ∈ Yλ . The simplest example of a q-multiple fibered surface is the Riemannian product of surfaces. In this case, any mapping ρλ in (12.1) depends only on the point xλ ∈ Xλ . There exist q-multiple fibered surfaces that cannot be represented as Riemannian products. For example, the Riemannian product of q two-dimensional spheres in three-dimensional Euclidean spaces is the configuration manifold of the mechanical system [4,130] consisting of q independent spatial pendulums. However, the configuration space of the mechanical system consisting of q consecutively connected spatial pendulums is a q-multiple fibered surface, which cannot be represented as a Riemannian product. A necessary condition of the infinitesimal rigidity of lth order, l = 1, 2, . . . , ∞ (the condition of analytic unbendability for l = ∞) yields the following assertion. Theorem 12.6. If a q-multiple fibered surface Fq possesses rigidity of order l ≥ 1 in the space Sq , then the base F1 possesses rigidity of order l in the space E1 . A point on a surface at which the dimension of the principal normal space does not coincide with the codimension is called an umbilic point. A sufficient condition of the infinitesimal rigidity of a q-multiple fibered surface Fq of class C 2 can be stated as follows. Theorem 12.7. Let the base F1 and any fiber Φyµ of a q-multiple fibered surface Fq of class C 2 contain no umbilic points and the type numbers of these surfaces satisfy the inequalities t(F1 ) ≥ 2 and t(Φyµ ) ≥ 2. Let the base F1 possess the (k, l)-rigidity, 1 ≤ k ≤ l ≤ ∞ (analytically unbendable for l = ∞) in the space E1 and any fiber Φyµ possess the (k, l)-rigidity in the space Eµ+1 . Then the surface Fq possesses the (k, l)-rigidity in the space Sq . Consider a q-multiple fibered surface Fq defined by a C 3 -immersion zq : Yq → Sq of the form (12.1), where λ
ρµλ ρλ = µ=1

C 3 -mappings,

and ρµλ : Xµ → Eλ are λ ≥ µ, such that ρλλ : Xλ → Eλ is a C 3 -immersion and mλ = nλ +1. Let Φλ = ρλλ (Xλ ). At an arbitrary point xλ ∈ Xλ , we consider a two-dimensional subspace T˜xλ (Xλ ) in the tangent space Txλ (Xλ ). At any point yq = (yλ−1 , xλ , xλ+1 , . . . , xq ) ∈ Yq , a two-dimensional subspace Tλ in the tangent space Tyq (Yq ) lying in the space Tyq ({yλ−1 } × Xλ × {(xλ+1 , . . . , xq )}) corresponds to T˜xλ (Xλ ). Assume that the following conditions hold: A. At any point of each of the manifolds Xλ , there exists a two-dimensional subspace T˜λ such that the curvature of the surface Φλ with respect to this subspace is nonzero. B. At any point yq ∈ Yq , for any λ = 1, . . . , q, on the two-dimensional subspace Tλ corresponding to T˜λ , the second fundamental forms II(νλ ) and II(νλ+1 ) of the surface Fq in Sq are not proportional. 886

Theorem 12.8 (see [93]). If a q-multiple fibered surface Fq satisfies conditions A and B and the base Φ1 possesses rigidity of order l ≥ 1 (analytically unbendable for l = ∞) in the space E1 , then the surface Fq possesses rigidity of order l in the space Sq . 12.4. Rigidity Theorems for Multidimensional Surfaces with Boundaries. In this section, we consider infinitesimal first-order bendings of a certain class of multidimensional surfaces with boundaries under slicing-type boundary conditions. Let E = E1 × · · · × En be a 3n-dimensional Euclidean space represented as the product of n three-dimensional Euclidean spaces E1 , . . . , En . In any space Ei , we fix an orthogonal Cartesian coordinate system Oxi y i z i , i = 1, . . . , n. We obtain the orthogonal Cartesian coordinate system Ox1 y 1 z 1 · · · xn y n z n in the space E. Denote by Πi a three-dimensional plane in E which is parallel to the coordinate plane Oxi y i z i and lies in the 3(n + 1 − i)-dimensional coordinate plane Oxi y i z i · · · xn y n z n . For fixed i < n, there exists a continuous family of planes Πi ; for i = n, such a plane is unique, and it coincides with the coordinate plane Oxn y n z n . We consider in E a 2n-dimensional surface F of class C 2 satisfying the following conditions.2 A . For any i = 1, . . . , n, the sections Φi of the surface F by the plane Πi , except, perhaps, for a finite number of such sections for i < n, is a two-dimensional convex surface of nonzero curvature and all sections Φ1 , . . . , Φn−1 are bounded. B . The surface F admits a bijective orthogonal projection on the coordinate plane Ox1 y 1 · · · xn y n . Condition B implies that the surface F is the image of a C 2 -immersion z : D → E of the form   z(u) = u, f 1 (u), . . . , f n (u) , where D is a domain in the plane Ox1 y 1 · · · xn y n , which is the projection of the surface F on this plane, and u = (x1 , y 1 , . . . , xn , y n ) is a point in D. Condition A implies that for i ≥ 2, the function f i in the coordinate representation of the immersion is independent of x1 , y 1 . . . , xi−1 , y i−1 . In particular, any section Φi is defined by the equation i+1 n n z i = f i (xi , y i , xi+1 0 , y0 , . . . , x0 , y0 ),

A

xk0 , y0k = const,

k > i.

B ,

and the following condition holds: Assume that, in addition to conditions  C . If the section Φn is not bounded, then the function f n (xn , y n ) has the form f n (xn , y n ) = ρ2 (xn , y n ) (1 + ε(xn , y n ))

as

|xn |, |y n | → ∞,

where ρ2 (xn , y n ) is a positive-definite quadratic form of the variables xn and y n and ε(xn , y n ) → 0. Under these conditions, we consider infinitesimal bendings of the surface F in the space E with bounded rotation fields and bounded variations of coefficients of the second fundamental forms. An infinitesimal bending of the surface F is called an infinitesimal gliding bending with respect to a given h-dimensional plane, 0 ≤ h ≤ 3n, if the bending field is parallel to this plane along the boundary ∂F (if h = 0, then the bending field is constant along the boundary). The following theorem is a higher-dimensional generalization of the classical theorem on the rigidity of a two-dimensional convex surface, which can be bijectively projected on a plane with respect to infinitesimal gliding bendings with respect to this plane (see, e.g., [150]). Theorem 12.9. If the projection D of a surface F satisfying conditions A –B on the plane Ox1 y 1 · · · xn y n is bounded, then the surface F does not admit nontrivial infinitesimal gliding bendings with respect to the 2n-dimensional plane Ox1 y 1 · · · xn y n . Note that in the multidimensional case, the dimension of the gliding plane plays the role of an additional condition on the bending field and imposes certain restrictions on it. The higher this dimension, the less strict the restrictions on the bending field that are imposed. For gliding planes of dimension 2n+1, the following theorem holds. 2

It can be proved that surfaces of this class are fibered surfaces.

887

Theorem 12.10. If the projection D of a surface F satisfying conditions A –B on the plane Ox1 y 1 · · ·xn y n is unbounded, then the surface F does not admit nontrivial infinitesimal gliding bendings with respect to the (2n + 1)-dimensional plane Ox1 y 1 · · · xn y n z n . This theorem is a higher-dimensional analogue of the theorem on the rigidity of an infinite, two-dimensional, convex surface in the three-dimensional space [113]. The proofs of Theorems 12.9 and 12.10 can be found in [91]. Applying the Darboux–Sauer theorem (see Sec. 11.4), one can replace condition B (the existence of a bijective projection on a plane) by the star-like condition with respect to some plane. A surface is said to be star-shaped with respect to an (h − 1)-dimensional plane Πh−1 if any open, h-dimensional half-plane with boundary Πh−1 has no more than one common point with this surface and does not touch this surface. If h = 1, then any plane Πh−1 is a point, and we obtain the definition of a star-shaped surface with respect to this point. Assume that instead of condition B , the considered 2n-dimensional surface F in the 3n-dimensional space E satisfies the following condition: B . The surface F is located in the half-space z n ≥ 0 and is star-shaped with respect to the (n − 1)-dimensional plane Πn−1 = Oz 1 · · · z n−1 , and the plane z n = 0 contains no boundary points of the surface F . Then the following theorem holds. Theorem 12.11. A surface F satisfying conditions A and B does not admit nontrivial infinitesimal gliding bendings with respect to the 2(n − 1)-dimensional plane Ox1 y 1 · · · xn−1 y n−1 in E. Since the gliding condition with respect to a given plane Π implies the gliding condition with respect to any plane of higher dimension containing the plane Π, Theorem 12.11 implies, in particular, that under conditions A and B , the surface F does not admit nontrivial infinitesimal bendings if its boundary is fixed. Conditions A and B imply that among all sections Φi , i = 1, . . . , n, by planes Πi , only Φn can be a closed surface. In this case, the following theorem holds. Theorem 12.12. If the section Φn is a closed surface, then the surface F satisfying conditions A and B does not admit nontrivial infinitesimal gliding bendings with respect to the (2n + 1)-dimensional plane Ox1 y 1 · · · xn y n z n . If the section Φn is not closed, then the dimension of the gliding plane in this theorem can be decreased. For example, the following theorem holds. Theorem 12.13. If the section Φn admits a bijective projection on the plane Oxn y n , then a surface F satisfying condition A and B does not admit nontrivial infinitesimal gliding bendings with respect to the 2n-dimensional plane Ox1 y 1 · · · xn y n . In the bibliography, we try to give a reference for each item to at least one of the following abstract journals: “Referativnyj Zhurnal ‘Matematika’” (RZhMat), “Zentralblatt MATH” (Zbl), and “Mathematical Reviews” (MR). REFERENCES 1. K. Abe and J. A. Erbacher, “Isometric immersions with the same Gauss map,” Math. Ann., 215, 197–201 (1975). (RZhMat, 1975, 12A704.) 2. C. B. Allendoerfer, “Rigidity for spaces of class greater than one,” Amer. Math. J., 61, No. 1, 633–644 (1939). (MR, 1940, Vol. 1, No. 1, p. 28; Zbl, 0021, 15803.) 3. Yu. A. Aminov, Geometry of Submanifolds [in Russian], Naukova Dumka, Kiev (2002). 4. V. I. Arnol’d, Mathematical Methods in Classical Mechanics, Grad. Texts Math., Vol. 60, Springer, New York (1978). (Zbl, 385, 70001; Zbl, 386, 70001.) 888

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