manuscripta math. 107, 1 – 13 (2002)

© Springer-Verlag 2002

Martin Svensson

On holomorphic harmonic morphisms Received: 28 May 2001 Abstract. We study holomorphic harmonic morphisms from Kähler manifolds to almost Hermitian manifolds. When the codomain is also Kähler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic.

1. Introduction A smooth map ϕ : (M, g) → (N, h) between two connected Riemannian manifolds is called a harmonic morphism if for any open subset U of N with ϕ −1 (U) nonempty and any harmonic function f on U, the composition f ◦ ϕ is harmonic on ϕ −1 (U). It was proved independently by Fuglede [4] and Ishihara [9] that a smooth map ϕ is a harmonic morphism if and only if ϕ is both a harmonic map and horizontally conformal. For a survey of the literature on harmonic morphisms see [6]. We shall here be concerned with holomorphic harmonic morphisms defined on a Kähler manifold with values in some almost Hermitian manifold. As is well known when the domain and the codomain are Kähler, holomorphicity implies harmonicity. So in that case, a holomorphic harmonic morphism is simply a holomorphic horizontally conformal map. Holomorphic harmonic morphisms from open subsets of Cm into Cn have been classified by Gudmundsson and Sigurdsson [8]. The case when n = 1 is very special; here the equations of harmonicity and horizontal conformality reduce to
∂ 2f =0 ∂zk ∂ z¯ k k

and


∂f ∂f = 0. ∂zk ∂ z¯ k k

Thus any holomorphic map from an open subset of Cm into C is a harmonic morphism. However, when n > 1 the class of holomorphic harmonic morphisms is M. Svensson: Centre for Mathematical Sciences, Mathematics, Faculty of Science, Lund University, Box 118, 221 00 Lund, Sweden. e-mail: [email protected]; [email protected] Mathematics Subject Classification (2000): 58E20

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M. Svensson

much more restrictive. According to their classification, a holomorphic harmonic morphism from an open subset of Cm into Cn with n > 1 is, up to composition with homothetic maps, just the standard projection Cm = Cn × Cm−n → Cn . This result is the main motivation for this paper and suggests that the class of holomorphic harmonic morphisms on a Kähler manifold should be very special. However, as a consequence of a result of Siu on harmonic maps, if M and N are compact Kähler manifolds where dimC N > 1 and N has strongly negative curvature, then any harmonic morphism from M to N is either holomorphic or anti-holomorphic [18]. A similar result holds for any harmonic morphism from a compact Kähler manifold to a compact quotient of some of the classical bounded symmetric domains. For the precise statement of Siu’s result, see [18, Theorem 3]. In Sect. 2 we prove a result which gives restrictions on holomorphic harmonic morphisms between Kähler manifolds where the domain has constant holomorphic curvature. Theorem 2.7. Suppose that M and N are Kähler manifolds with dimC N > 1 and ϕ : M → N is a non-constant holomorphic harmonic morphism. If M has constant holomorphic curvature then so has the open submanifold ϕ(M) of N . If in addition M is compact, then ϕ has constant dilation and the holomorphic sectional curvatures of M and N differ only by a multiple given by a positive real number. As two new examples of maps of this kind, we study the projection [z0 , . . . , zm ] → [z0 , . . . , zn ] from an open subset of CP m to CP n and a similar projection from CH m to CH n . In Sect. 3 we derive a Bochner formula for holomorphic harmonic morphisms. Using this we prove the following result. Theorem 3.3. If M is a compact Kähler manifold of non-negative sectional curvature and N is an almost Hermitian manifold, then the regular fibres of a holomorphic harmonic morphism from M to N are totally geodesic submanifolds of M. Applying this to Kähler submersions, i.e. holomorphic Riemannian submersions between Kähler manifolds, gives us the following main result. Theorem 3.4. If M is a Kähler manifold of non-negative sectional curvature, then any Kähler submersion on M is a totally geodesic map. Using our Bochner formula we also produce a new proof of the above mentioned classification of Gudmundsson and Sigurdsson.

2. Holomorphic harmonic morphisms Suppose (M, g) and (N, h) are Riemannian manifolds and ϕ is a harmonic morphism from M to N . By the earlier mentioned result of Fuglede and Ishihara this

On holomorphic harmonic morphisms

3

means that ϕ is both harmonic and horizontally (weakly) conformal. That ϕ is harmonic means that trace ∇dϕ = 0,

(2.1)

where dϕ is considered as a section of the bundle T ∗ M ⊗ ϕ −1 T N with its natural connection. In local coordinates this gives a second order semi-linear elliptic system of partial differential equations for ϕ to satisfy. That ϕ is horizontally (weakly) conformal means that at each point x ∈ M, either dϕx = 0 or the restriction of dϕx onto the orthogonal complement of its kernel in Tx M is surjective and conformal. However, if ϕ is also harmonic, we can only have dϕx = 0 for x in a nowhere dense subset of M unless ϕ is constant. This follows from an application of Aronszajn’s uniqueness theorem to the harmonicity equation (2.1) above, see [16]. Thus on an open dense subset of M we have two orthogonal distributions, the vertical V and the horizontal H, where Vx = ker dϕx . Moreover, there is a positive function λ defined on this dense open subset with the property that h(dϕx (X), dϕx (Y )) = λ(x)2 g(X, Y ),

X, Y ∈ H.

We extend λ to the whole of M by demanding it to be zero at points where dϕ vanishes; this extended function we call the dilation of ϕ. A horizontally conformal map is called horizontally homothetic if the gradient of its dilation is vertical at every regular point, i.e. at every point where the differential of ϕ is non-vanishing. Horizontally conformal maps generalize the well-known Riemannian submersions which have been studied extensively. To avoid cumbersome notation we will from now on suppress the metric when referring to a Riemannian manifold; in calculations we shall simply write  , . We denote by J the complex structure of the specific manifold and unless otherwise specified, by the dimension of a manifold we always mean the real dimension. All the manifolds and maps appearing are assumed to be smooth and all the manifolds are assumed to be connected. The following well-known results will be needed. Proposition 2.1 ([5]). A horizontally homothetic map has no critical points of finite order. In particular, any non-constant horizontally homothetic harmonic map is submersive. Proposition 2.2 ([1, 2]). Let ϕ : M → N be a horizontally conformal map between two Riemannian manifolds M and N . (a) If dim N = 2, then ϕ is harmonic if and only if its fibres are minimal at regular points. (b) If dim N > 2 then any two of the following assertions imply the third: (i) ϕ is harmonic, (ii) ϕ has minimal fibres at regular points, (iii) ϕ is horizontally homothetic.

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Remark 2.3. If M is a Kähler manifold, N is an almost Hermitian manifold and ϕ : M → N a non-constant holomorphic harmonic morphism, then the fibres of ϕ at regular points will be complex submanifolds of M and thus minimal by a well known result (see e.g. [19, Theorem 1.5]). If dim N > 2, then it follows from Proposition 2.2 that ϕ is horizontally homothetic and moreover submersive by Proposition 2.1. We shall also make use of O’Neill’s fundamental tensors A and T defined in [14] by AE F = V(∇HE HF ) + H(∇HE VF ), TE F = V(∇V E HF ) + H(∇V E VF ), for vector fields E and F . Lemma 2.4. Let M be a Kähler manifold and N an almost Hermitian manifold. If ϕ : M → N is a horizontally conformal holomorphic submersion with dilation λ, then V[X, Y ] = −λ2 J X, Y J V∇λ−2 , B H (X, Y ) = −

λ2 X, Y V∇λ−2 , 2

for horizontal vector fields X and Y . Proof. Recall that for horizontal vector fields X and Y we have [7, Proposition 2.1.2] AX Y =

 1 V[X, Y ] − λ2 X, Y V∇λ−2 . 2

(2.2)

When proving the first identity, we may without loss of generality assume that X and Y are basic vector fields, i.e. horizontal vector fields which project by dϕ to vector fields on N . Then [V , X] is vertical for any vertical vector field V and J X is also basic. From Eq. (2.2) we see that V∇X J X = J V∇X X implies V[X, J X] = −λ−2 X, XJ V∇λ−2 .

(2.3)

Next we show that V[X, J Y ] = V[Y, J X]. Using that X and Y are basic we get if V is a vertical vector field [X, J Y ], V  = ∇X J Y, V  − ∇J Y X, V  = −J Y, ∇X V  + X, ∇J Y V  = −J Y, ∇V X + X, ∇V J Y  = Y, ∇V J X − J X, ∇V Y  = Y, ∇J X V  − J X, ∇Y V  = −∇J X Y, V  + ∇Y J X, V  = [Y, J X], V .

On holomorphic harmonic morphisms

5

Thus by symmetry of both sides of equation (2.3) we see that V[X, J Y ] = −λ−2 X, Y J V∇λ−2 . By replacing Y with J Y we thus get the first of the identities of the lemma. To prove the second identity, just note that B H (X, Y ) =

1 λ2 (AX Y + AY X) = − X, Y V∇λ−2 , 2 2

by equation (2.2). The lemma is proven.

 

Proposition 2.5. Let M be a Kähler manifold and N be an almost Hermitian manifold. If ϕ : M → N is a horizontally conformal holomorphic submersion, then the following assertions are equivalent: (i) the horizontal distribution is integrable, (ii) the horizontal distribution is totally geodesic, (iii) the gradient of the dilation of ϕ is horizontal. Proof. This follows immediately from Lemma 2.4.   Remark 2.6. The above conditions are all equivalent to the vertical distribution being Riemannian, see e.g. [12]. Remark 2.7. If ϕ is a holomorphic harmonic morphism and dim N > 2 then the conditions of Proposition 2.5 are all equivalent to the fact that the dilation of ϕ is constant. If ϕ : M → N is a horizontally homothetic submersion with dilation λ, then we have the following formula of Gudmundsson [7, Theorem 2.2.5]: 3 λ4 K M (X ∧ Y ) = λ2 K N (ϕ∗ X ∧ ϕ∗ Y ) − |V[X, Y ]|2 − |∇λ−2 |2 , 4 4

(2.4)

which holds for horizontal orthonormal vectors X and Y on M. Here K M and K N are the sectional curvatures of the corresponding 2-planes on M and N , respectively. This may be simplified in the case of holomorphic sectional curvature as the following lemma shows. Lemma 2.8. Let M be Kähler, N almost Hermitian and ϕ : M → N a horizontally homothetic holomorphic map with dilation λ, then for a horizontal unit vector X on M: K M (X ∧ J X) = λ2 K N (ϕ∗ X ∧ J ϕ∗ X) − λ−4 |∇λ2 |2 .

(2.5)

In particular, this holds when ϕ is a holomorphic harmonic morphism and N is of complex dimension at least 2. Proof. This follows immediately from Eq. (2.4) and Lemma 2.4.

 

From Lemma 2.8 we get restrictions on holomorphic harmonic morphisms defined on Kähler manifolds of constant holomorphic curvature.

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Theorem 2.9. Suppose that M and N are Kähler manifolds with dimC N > 1 and ϕ : M → N is a non-constant holomorphic harmonic morphism. If M has constant holomorphic curvature then so has the open submanifold ϕ(M) of N . If in addition M is compact, then ϕ has constant dilation and the holomorphic sectional curvatures of M and N differ only by a multiple of a positive real number. Proof. If α M is the constant holomorphic curvature of M, then for a horizontal unit vector field X on M we have by Eq. (2.5) α M = λ2 K N (ϕ∗ X ∧ J ϕ∗ X) − λ−4 |∇λ2 |2 .

(2.6)

Hence the holomorphic curvature of a complex line at a point of N only depends on the point, not on the chosen line. By a well known Kählerian analogue of Schur’s theorem (see e.g. [11, Theorem 7.5]), N has constant holomorphic sectional curvature. If we denote this constant by α N we have by Eq. (2.6) α M = λ2 α N − λ−4 |∇λ2 |2 .

(2.7)

If M is compact then λ2 attains a minimum and a maximum at points of M where its gradient vanishes. Thus if α N = 0 then we must have α M = 0 as well. If on the other hand α N  = 0, then by equation (2.7) the values of λ2 at the extremal points are the same. Hence λ is constant and equation (2.7) reads α M = λ2 α N . This proves the last assertion.   We give two examples of holomorphic harmonic morphisms between Kähler manifolds of constant holomorphic curvature. Example 2.10. We define the following map ϕ : U ⊆ CP m → CP n ,

ϕ([z0 , . . . , zm ]) = [z0 , . . . , zn ],

where U = {[z] ∈ CP m |zk  = 0 for some k ≤ n}. Clearly, ϕ is holomorphic, and we shall prove that ϕ is horizontally conformal. Take a point [z] ∈ U . We may assume that z0  = 0, and thus apply local coordinates around [z] and ϕ([z]). Thus we get a map ϕ : C m → Cn , which is just orthogonal projection onto the first n variables. The corresponding Hermitian metric which we have in Cm is   4 X, Y z = (1 + |z|2 )X, Y  − X, zz, Y  . 2 2 (1 + |z| ) For ξ ∈ Cm we write ξ = ξ  + ξ  where ξ  ∈ Cn × {0}. Thus we see that the kernel of dϕz is {η = η }. If η = η then   4 γ , z    , − z , η γ γ , ηz = (1 + |z|2 ) 1 + |z|2

On holomorphic harmonic morphisms

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so that the horizontal space at z is given by  γ , z   z . Hz = γ | γ  = 1 + |z|2 Thus for γ ∈ Hz we get |γ , z|2  2 |z | (1 + |z|2 )2 1 + |z |2 γ  , z  = γ , z . 1 + |z|2 |γ  |2 = |γ |2 −

From this follows that |dϕz (γ )|2z =

1 + |z|2 |γ |2 , 1 + |z |2 z

so ϕ is horizontally conformal with dilation λ(z)2 =

1 + |z|2 . 1 + |z |2

In particular, λ is not constant along vertical curves so H is not integrable according to Proposition 2.5. For a point [z] = [z0 , . . . , zn ] in CP n , define the subspace V = C · (z0 , . . . , zn ) × Cm−n of Cm+1 . If we consider CP m as the quotient space of Cm+1 \ {0} under the action of C∗ , the fiber of ϕ over [z] is the quotient of V intersected with the domain U . From this it is clear that ϕ has totally geodesic fibres. Example 2.11. Consider the open unit ball in Cm as a model for the complex hyperbolic space CH m . With this model we have the Bergman metric (see [11, p. 162], where there is a misprint) X, Y z =

  4 (1 − |z|2 )X, Y  + X, zz, Y  . 2 2 (1 − |z| )

On the right-hand side we have used the usual Hermitian product in Cm . Next let m > n and consider the map ϕ : CH m → CH n ,

ϕ(z1 , . . . , zm ) = (z1 , . . . , zn ) =: (z , 0),

where we use the same notation as in the previous example. Clearly, ϕ is holomorphic. For a point z ∈ CH m the kernel of dϕz may be written as {η ∈ Cm |η = η }. Thus if η = η , we get   4 (1 − |z|2 )γ  , η  + γ , zz , η  2 2 (1 − |z| )   4 γ , z    = , + , η γ z (1 − |z|2 ) (1 − |z|2 )

γ , ηz =

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so that  Hz = γ ∈ Cm |γ  +

 γ , z  = 0 . z (1 − |z|2 )

If γ is horizontal at z: |γ , z|2  2 |z | , (1 − |z|2 )2 γ , z γ , z = γ  , z  + γ  , z  = γ  , z  − |z |2 , (1 − |z|2 ) |γ |2 = |γ  |2 + |γ  |2 = |γ  |2 +

and the last of these equations implies that  1 − |z |2 |z |2 γ  , z  = γ , z 1 + = γ , z . 2 1 − |z| 1 − |z|2 Putting these together we get for γ ∈ Hz :   4 (1 − |z|2 )|γ  |2 + |γ  , z |2  2 2 (1 − |z | )  4 |γ , z|2  2  2 2 = (1 − |z | )(|γ | − |z | ) (1 − |z |2 )2 (1 − |z|2 )2 (1 − |z |2 )2 + |γ , z|2 (1 − |z|2 )2  4 = (1 − |z |2 )|γ |2 (1 − |z |2 )2 (1 − |z |2 )2 (1 − |z |2 )|z |2 + |γ , z|2 ( − ) . (1 − |z|2 )2 (1 − |z|2 )2

|dϕz (γ )|2 = γ  , γ  z =

We see that (1 − |z |2 )2 − (1 − |z |2 )|z |2 (1 − |z |2 2 = (1 − |z − |z |2 ) (1 − |z|2 )2 (1 − |z|2 )2 1 − |z |2 = . 1 − |z|2 Using this in the above calculation gives   2 4 2  2 2 2 1 − |z | |dϕz (γ )| =  2 (1 − |z | )|γ | + |γ , z| 1 − |z|2 1 − |z |2

 4 1 − |z|2 2 2 2 )|γ | + |γ , z| (1 − |z| =   1 − |z |2 1 − |z|2 2 =

1 − |z|2 γ , γ z . 1 − |z |2

On holomorphic harmonic morphisms

9

Thus we have proved that ϕ is horizontally conformal with dilation λ(z)2 =

1 − |z|2 . 1 − |z |2

Since holomorphic, ϕ is harmonic and thus a harmonic morphism. Note that λ is not constant along vertical curves, and by Proposition 2.5 we see that H is not integrable. As in the previous example, it it easy to see that ϕ has totally geodesic fibres. 3. A Bochner formula In this section we prove a Bochner-type formula for holomorphic harmonic morphisms defined on Kähler manifolds and deduce from this the main results of this paper. Recall that for a Riemannian vector bundle E with a Hermitian structure J in each fibre, a complex orthonormal frame of E is simply a local orthonormal frame {ξ1 , . . . , ξ2m } of E where J ξ2k−1 = ξ2k for k = 1, . . . , m. Proposition 3.1. If M is a Kähler manifold and N is an almost Hermitian manifold, then for a submersive holomorphic harmonic morphism from M to N we have
K M (ei ∧ eα ) + B V 2 = divV trace B H . (3.1) i,α

Here {ei } is a local complex orthonormal frame for the vertical distribution V, {eα } is a local complex orthonormal frame for the horizontal distribution H and B V 2 denotes the Hilbert–Schmidt norm of B V . We will prove Proposition 3.1 at the end of this section. In this context, the following result ought to be mentioned. Proposition 3.2. If M is a compact Kähler manifold of positive sectional curvature, then there are no non-constant holomorphic maps from M into any Kähler manifold of strictly lower dimension. Proof. Since the sectional curvature is positive, h1,1 (M) = 1 by a result of Bishop and Goldberg [3]. The proposition now follows from a result of Sampson [17, p. 98].   Theorem 3.3. If M is a compact Kähler manifold of non-negative sectional curvature and N is an almost Hermitian manifold, then the fibres over regular values of a holomorphic harmonic morphism from M to N are totally geodesic submanifolds of M. Proof of Theorem 3.3. If ϕ : M → N is a holomorphic harmonic morphism and x ∈ N a regular value of ϕ, then the dilation of ϕ is non-zero in a neighbourhood U ⊂ M of the fibre of ϕ over x. Hence ϕ is submersive in U and the fibre over x is a compactly embedded submanifold of U . Integrating equation (3.1) over the fibre gives the result.   When the dilation is constant we can say even more.

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Theorem 3.4. If M is a Kähler manifold of non-negative sectional curvature, then any Kähler submersion on M is a totally geodesic map. Proof of Theorem 3.4. Since the dilation of such a map is constant, the horizontal distribution is integrable by Proposition 2.5 and the fibres are also totally geodesic by Proposition 3.1. By a well-known characterization of totally geodesic horizontally conformal maps due to Mustafa [13, Theorem 2.3], the map is totally geodesic.   The Bochner formula of Eq. (3.1) also furnishes another proof of the classification of Gudmundsson and Sigurdsson of holomorphic harmonic morphisms between Euclidean spaces. Theorem 3.5 ([8]). If ϕ : U ⊂ Cm → Cn is a holomorphic harmonic morphism with U open and n > 1, then ϕ is an orthogonal projection followed by a homothety (i.e a conformal map which preserves the metric up to a constant factor). Proof of Theorem 3.5. The map ϕ is horizontally homothetic by Proposition 2.1 so Eq. (2.4) applies. However, due to flatness of the spaces all terms in the formula must vanish. This implies that the dilation is constant so the map ϕ is totally geodesic by Theorem 3.4. Hence it has to be an affine map, and being horizontally conformal, it has to be an orthogonal projection onto the horizontal space followed by a homothety.   For the proof of Proposition 3.1 we need some preliminary results. We state and prove them under more general conditions than the result requires. Lemma 3.6. Suppose that M is a Kähler manifold endowed with an integrable complex distribution V and denote H = V ⊥ . If V is a section of V and X is a section of H, then V(∇J V X + ∇V J X) = 0, where V also denotes orthogonal projection onto V. Remark 3.7. As the proof shows, the condition on M to be Kähler can be weakened. It is enough for M to be almost Hermitian and V superminimal, i.e. ∇V J = 0 for every section V of V. Proof. The distribution V is integrable so for any section W of V we get ∇J V X, W  = −X, ∇J V W  = −X, ∇W J V  = J X, ∇W V  = J X, ∇V W  = −∇V J X, W , which proves the lemma.  

On holomorphic harmonic morphisms

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We may now apply Lemma 3.6 to prove the following: Lemma 3.8. Assume that M is a Kähler manifold endowed with an integrable complex distribution V and denote H = V ⊥ . Let {ei } be a local complex orthonormal frame for V and X any section of H. Then
(∇X T )ei X, ei  = 0. i

Proof. By definition we have (∇X T )ei X, ei  = ∇X (Tei X), ei  − T∇X ei X, ei  − Tei ∇X X, ei  = ∇X V(∇ei X), ei  − ∇V (∇X ei ) X, ei  − ∇ei H(∇X X), ei . When summing over i the last term vanishes since

∇ei H(∇X X), ei  = − (H(∇X X), ∇ei ei  = 0, i

i

due to the minimality of the leaves of V. As for the other two terms we have from Lemma 3.6 and the Kähler condition: ∇X V(∇J ei X), J ei  − ∇V (∇X J ei ) X, J ei  = −∇X V(∇ei J X), J ei  + ∇V (∇X ei ) J X, J ei  = −∇X V(∇ei X), ei  + ∇V (∇X ei ) X, ei . Thus when summing over a complex vertical frame {ei , J ei } the terms will cancel out. Lemma 3.9. Assume that M is a Kähler manifold endowed with an integrable complex distribution V and denote H = V ⊥ . If V is a section of V and X is a section of H, then H∇X J V = H∇J X V . Proof of Lemma 3.9. Extend X to a basic vector field. Then H∇X J V = J H∇X V = J H∇V X = H∇V J X = H∇J X V . This gives the result.   Proof of Proposition 3.1. We start out with a formula of Ranjan on the mixed curvatures [15, page 87]: R M (V , X)Y, W  = (∇V A)X Y, W  + (∇X T )V W, Y  − TV X, TW Y  − AAX V W, Y ,

(3.2)

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Fix a point p ∈ M and take local orthonormal complex frames {ei } for V and {eα } for H around p. Thus we get by summing over i and α in (3.2):


K M (eα ∧ ei ) = (∇ei A)eα eα , ei  − AAeα ei ei , eα  i,α

i,α

i,α



+ (∇eα T )ei ei , eα  − |Tei eα |2 . i,α

(3.3)

i,α

According to Lemma 3.8 the third sum vanishes. As for the second sum, since we are summing over complex frames for V and H, we get by definition:


AAeα ei ei , eα  = ∇H∇eα ei ei , eα  + ∇H∇J eα ei ei , J eα  i,α

i,α

i,α


+ ∇H∇eα J ei J ei , eα  i,α

+


i,α

(3.4)

∇H∇J eα J ei J ei , J eα .

Due to Lemma 3.9, the last sum cancels out the first and the third cancels out the second in Eq. (3.4). Finally, as for the first term in Eq. (3.3), note that 1 1 A(∇ei eβ ) eβ = V[H(∇ei eβ ), eβ ] = − V[eβ , H(∇ei eβ )] = −Aeβ (∇ei eβ ). 2 2 Hence ∇ei (Aeβ eβ ) = (∇ei A)eβ eβ . This completes the proof of Proposition 3.1.

 

Acknowledgements. I wish to express my gratitude to Dr. Sigmundur Gudmundsson and Prof. John C. Wood for their helpful comments and advice.

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