Horizontal Laplace Operator in Real Finsler Vector Bundles

Acta Mathematica Scientia 2008,28B(1):128–140 http://actams.wipm.ac.cn

HORIZONTAL LAPLACE OPERATOR IN REAL FINSLER VECTOR BUNDLES∗ 1,2

 )

Zhong Chunping (

1 School

1


)

Zhong Tongde (

of Mathematical Sciences, Xiamen University, Xiamen 361005, China 2 Department of Mathematics, Zhejiang University, Hangzhou 310028, China E-mail: [email protected]

Abstract A vector bundle F over the tangent bundle T M of a manifold M is said to be a Finsler vector bundle if it is isomorphic to the pull-back π ∗ E of a vector bundle E over M ([1]). In this article the authors study the h-Laplace operator in Finsler vector bundles. An h-Laplace operator is defined, first for functions and then for horizontal Finsler forms on E. Using the h-Laplace operator, the authors define the h-harmonic function and hharmonic horizontal Finsler vector fields, and furthermore prove some integral formulas for the h-Laplace operator, horizontal Finsler vector fields, and scalar fields on E. Key words h-Laplace operator, h-harmonic, Finsler vector bundle 2000 MR Subject Classification

1

53C60, 53C65, 58A14

Introduction

It is well known that various kinds of Laplace operators play a very important role in differential geometry and physics, especially in the theory of harmonic integral and Bochner technique. Using Bochner technique, [2] proved some theorems on non-existence of certain types of vector fields (such as Killing vector fields and conformal Killing vector fields) on a compact Riemannian manifold whose Ricci curvature is positive or negative definite. Local expressions of any Laplace operators are called Weitzenb¨ock formulas. These formulas primarily show the application of Bochner technique. Weitzenb¨ock formulas of the Laplace operator for forms on Riemannian manifolds inspired intensive and important researches (see [3] and references therein). The key point in the theory of harmonic integrals and Bochner technique is to define a suitable Laplace operator, harmonic functions, and forms. The purpose of this article is to generalize the Laplace operator in Riemannian manifolds to Finsler vector bundles as such bundles arise naturally in Finsler geometry [4–6]. We define h-Laplace operator first for functions and then for horizontal Finsler forms on E. We define the h-harmonic function and h-harmonic horizontal Finsler vector fields and using the h-Laplace ∗ Received

December 11, 2005. This work was supported by Tian Yuan Foundation of China (10526033), China Postdoctoral Science Foundation (2005038639), and the Natural Science Foundation of China (10601040, 10571144).

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operator, we prove some integral formulas of the scalar fields and horizontal Finsler vector fields on E. In the end we obtain the Weitzenb¨ock formula of the h-Laplace operator for the horizontal Finsler forms on E.

2

Finsler Vector Bundles and Finsler Connections

In this section we introduce some notations and definitions [1]. Let us consider a compact manifold M , dimM = m, {U, (xi )} the coordinates in a local chart, and let π : E → M be a vector bundle E of rank r over M with compact fibre π −1 (x). Definition 2.1 [1] A vector bundle F over the tangent bundle T M of M is said to be a Finsler vector bundle if it is isomorphic to the pull-back π ∗ E. Definition 2.2 [1] A Finsler structure g on F is a smooth field of inner products in the fibres of F. Note that the natural projection, π, defines on the vertical bundle, V = {X ∈ T E|dπ(X) = 0} a structure of vector bundle of rank r over E. We denote by XV (E) the module of its sections, called vertical vector fields. The vertical vector fields form a subalgebra, XV (E), of the Lie algebra, X (E), and is a finitely generated module over the ring of the smooth functions C ∞ (E) of E. A given supplementary subbundle H of V, called horizontal bundle, that is, TE = H ⊕ V

(2.1)

defines a non-linear connection on E, and we denote by XH (E) the module of its sections, called horizontal Finsler vector fields. The horizontal Finsler vector fields on E form a finitely generated projective module XH (E) over C ∞ (E). However, they do not, in general, form a subalgebra of the Lie algebra X (E). It is a result of [1] that the horizontal bundle H and the vertical bundle V are Finsler vector bundles. Let {U, (xi , y a )} be the local coordinates on E and put ∂i = ∂/∂xi , ∂˙a = ∂/∂y a . In this article, the indices i, j, k, l, · · · take the values 1, · · · , m, and a, b, c, d, · · · take the values 1, · · · , r, and the usual summation convention for repeated indices is used. If Nia (x, y) are the coefficients of the non-linear connection, then the following vector fields, {δi = ∂i − Nia ∂˙a }, {∂˙a }, are called the local adapted bases of H and V, respectively. Their dual adapted bases are denoted by {dxi }, {δy a = dy a + Nia dxi }. For every vector field X on E, there exists a unique decomposition X = X H + X V , X H ∈ H, X V ∈ V. X H is called the horizontal part and X V the vertical part of X. m+r P p Let A(E) = ∧ (E) be the space of smooth forms on E. We denote by H∗ the dual p=0

vector bundle of H. Then a horizontal 1-form or horizontal Finsler 1-form is a smooth section

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of H∗ . If ω ∈ ∧1 (E) is a Finsler 1-form field on E, then it has a unique decomposition ω = ωH + ωV . ω H is called the horizontal component of ω and ω V the vertical component of ω. Note that ω H (X V ) = ω V (X H ) = 0. The horizontal Finsler forms are a graded subalgebra of A(E). This algebra is called the horizontal subalgebra, and is denoted by AH (E). A Finsler connection on E is a linear connection ∇ on E, which is both horizontal and vertical metrical. The existence of such Finsler connection on E was proved in [4]. If ∇ is a Finsler connection on E, we have ∇X Y = (∇X Y H )H + (∇X Y V )V , ∀X, Y ∈ X (E), ∇X ω = (∇X ω H )H + (∇X ω V )V , ∀X ∈ X (E), ω ∈ X ∗ (E). If we put V ∇H X Y = ∇X H Y and ∇X Y = ∇X V Y, ∀X, Y ∈ X (E),

then ∇ = ∇H + ∇V , where ∇H is called the h-covariant derivative and ∇V is called the v-covariant derivative of the Finsler connection. i a Locally there exists a well-determinded set of differentiable functions {Fjk (x, y), Fbk (x, y), i a −1 Cja (x, y), Cbc (x, y)} on π (U ) such that i a ˙ i a ˙ ∇δk δj = Fjk ∂a , ∇∂˙c δj = Cjc ∂a . δi , ∇δk ∂˙b = Fbk δi , ∇∂˙c ∂˙b = Cbc

Definition 2.3 A Finsler tensor field T on E is called a horizontal Finsler covariant tensor field of type p if T (X1 , · · · , Xp ) = T (X1H , · · · , XpH ), where Xi ∈ X (E), and XiH denote the horizontal part of Xi , etc. A horizontal differential form of type p or horizontal Finsler p-form over −1 π (U ) ⊂ E is a differentiable section, T : (x, y) → [(x, y), Ti1 ···ip (x, y)], of ⊗p H∗ such that the fibre coordinates, Ti1 ···ip (x, y), are skew-symmetric with respect Under the coordinate transformation on E, Ti1 ···ip (x, y) changes according to the Ti′1 ···i′p (x′ , y ′ ) = Ti1 ···ip (x, y)

∂xi1 ′ ′ i1

∂x

···

an open set over π −1 (U ) to i1 , · · · , ip . law:

∂xip ′ . ∂x′ ip

Similarly, we can define the horizontal Finsler contravariant tensor field of type p on E. If T i1 ···ip (x, y) is a horizontal Finsler contravariant tensor, then it changes according to the law: i′

T

E.

i′1 ···i′p

′

′

(x , y ) = T

i1 ···ip

i′

∂x′ 1 ∂x′ p (x, y) i1 · · · . ∂x ∂xip

In the following, we denote, by ∧pH (E), the space of smooth horizontal Finsler p-forms on

The decomposition (2.1) induces two projections, p∗H and p∗V , of df onto the horizontal Finsler forms and vertical Finsler forms, respectively. Set dH = p∗H ◦ d, dV = p∗V ◦ d.

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Here dH f = δi (f )dxi

and dV f = ∂˙a (f )δy a , f ∈ C ∞ (E).

So that the differential operator, d, can be decomposed as d = dH +dV when it acts on horizontal finsler forms. Notice that if ϕ= we have dH ϕ =

1 ϕi ···i dxi1 ∧ · · · ∧ dxip ∈ ∧pH (E), p! 1 p

1 δk (ϕi1 ···ip )dxk ∧ dxi1 ∧ · · · ∧ dxip ∈ ∧p+1 H (E). p!

Furthermore, we have dH (ϕ ∧ ψ) = dH ϕ ∧ ψ + (−1)p ϕ ∧ dH ψ, ϕ ∈ ∧pH (E), ψ ∈ ∧qH (E). Therefore, dH restricted to the horizontal Finsler forms is a type preserving differential operator. In this article, we are mainly interested in the horizontal bundle, H, and the h-covariant derivative, ∇H , of the metrical Finsler connection ∇. In the following, we restrict our discussion on horizontal bundle, H, its dual bundle, H∗ , and their tensor bundles. Now we shall investigate the h-covariant derivatives of tensors of arbitrary rank by means of the metrical connection. For a horizontal Finsler contravariant tensor field T i1 ···ip (x, y), its horizontal covariant derivative is p X iσ ∇δi T i1 ···ip = δi (T i1 ···ip ) + T i1 ···iσ−1 kiσ+1 ···ip Fki . (2.2) σ=1

For a horizontal Finsler covariant tensor of type Ti1 ···ip (x, y), the corresponding horizontal covariant derivative is given by ∇δi Ti1 ···ip = δi (Ti1 ···ip ) −

p X

Ti1 ···iσ−1 kiσ+1 ···ip Fikσ i .

(2.3)

σ=1

By means of the above formulas it is natural to extend the horizontal covariant derivatives of tensor of mixed covariant -and contra-variant valencies.

3

Horizontal Laplace Operator for Functions on E

In this section we assume that there is a Riemannian structure, G, on the total space, E, of a vector bundle, π : E → M . If V is the vertical space, then the vectors orthogonal to V, with respect to G, uniquely determine the horizontal space, H, and the map, N : (x, y) ∈ E → H defines a non-linear connection, N , on the total space E. A Finsler connection, ∇ on E, which preserves by parallelism the distributions, H and V, is called compatible with the Riemannian structure, G, if ∇X G = 0, ∀X ∈ X (E). By Proposition 4.1 and 4.2 in [4], there exist two positive symmetric Finsler tensor field, gij (x, y) and Gab (x, y), defined on E such that in the adapted local frame {δi , ∂˙a }, G(x, y) = gij (x, y)dxi ⊗ dxj + Gab (x, y)δy a ⊗ δy b .

(3.1)

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In fact gij (x, y) = G(δi , δj ), Gab (x, y) = G(∂˙a , ∂˙b ). Example 3.1 Let (M, F ) be a real Finsler manifold. Then the tangent bundle, T M , endowed with the Sasaki-type metric constructed from the given Finsler metric, F , is a Riemannian vector bundle and we have gij (x, y) = Gij (x, y). ∂ In the above case, as in [7], we denote by C = y i ∂y i the Liouville vector field, E = i j i gij (x, y)y y the absolute energy of (M, F ), and S = y δi the geodesic spray of (M, F ). If F comes from a Riemannian metric, namely, F (x, y) = gij (x)y i y j , then the Riemannian structure on T M can be chosen to be the Sasaki metric[8]: G(x, y) = gij (x)dxi ⊗ dxj + gij (x)δy i ⊗ δy j , Nji = Γimj (x)y m ,

(3.2)

and the geometry of T M with this metric was investigated in [8]. In the case that (M, F ) is a Finsler manifold, we denote i Pjk :=

∂Nji i − Fjk . ∂y k

(3.3)

i If Pjk = 0,then we call (M, F ) a Landsberg space [9]. Obviously, if (M, F ) is a Riemannian i i manifold then by (3.2) and (3.3) we have Pjk = 0. The further meaning of the vanishing of Pjk was investigated in [10]. In the following, we will see that the horizontal Laplace operator has a simple form both for scalar fields and for horizontal Finsler forms in Landsberg spaces. Now we denote by g jk and G bc the inverse matrix components of (gij ) and (Gab ), respeci a i a tively. Then the coefficients (Fjk , Fbk , Cjb , Cbc ) of the metrical Finsler connection, ∇, determined by the structure, G, are given by

1 ih ∂Nka a g (δk ghj + δj ghk − δh gjk ), Fbk (x, y) = , 2 ∂y b 1 i a Cjb (x, y) = 0, and Cbc = G ad (∂˙c Gdb + ∂˙b Gdc − ∂˙d Gbc ). 2

i (x, y) = Fjk

(3.4)

Notice that the h(v)-metrical of the Finsler connection gives ∇δi gjk = ∇δi g jk = 0

and ∇∂˙a Gbc = ∇∂˙a G bc = 0.

(3.5)

Since gij and Gab are two symmetric Finsler tensors, it follows from (3.4) that i i Fjk = Fkj

c c and Cab = Cba .

(3.6)

Now if we denote g = det(gij ) and G = det(Gab ), then, like in Riemannian case, it is easy to check that √ √ 1 1 √ √ k b Fik = √ δi ( g) = δi (ln g) and Cab = √ ∂˙a ( G) = ∂˙a (ln G). g G

(3.7)

In the following we will repeatedly use (3.5) and (3.7) without explicit statement. Now we define the h(v)-Laplace operator for a scalar field, f ∈ C ∞ (E). Notice that df = dH (f )dxi + dV (f )δy a

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for any scalar field, f ∈ C ∞ (E), and furthermore, we have dV (f ) = 0 if and only if f ∈ C ∞ (E) or f ∈ π ∗ C ∞ (M ). Thus, we are mainly interested in the horizontal derivative operator dH . √ √ Let dV = g Gdx1 ∧ . . . ∧ dxm ∧ δy 1 ∧ · · · ∧ δy r be the volume form associated to the Riemannian structure, G, on E and LX be the Lie derivative with respect to X ∈ X (E), then as in [7], the notations as gradient and divergent on E can be introduced. The divergence of X = X i δi + X˙ a ∂˙a ∈ X (E) is defined by LX dV = (divX)dV.

(3.8)

If we denote divh X =: div(X i δi ) and divv X =: div(X˙ a ∂˙a ), then we have Proposition 3.1 Let X = X i δi + X˙ a ∂˙a ∈ X (E), then we have divX = divh X + divv X,

(3.9)

where divh X = ∇δi X i − Pi X i and

and divv X = ∇∂˙a X˙ a + P˙a X˙ a ,

√ √ Pi = ∂˙a (Nia ) − δi (ln G) and P˙a = ∂˙a (ln g).

(3.10)

(3.11)

Proof Notice that [δi , δj ] = [δj (Nia ) − δi (Nja )]∂˙a , [δi , ∂˙a ] = ∂˙a (Nib )∂˙b

and [∂˙a , ∂˙b ] = 0,

and [X, δi ] = −δi (X j )δj + X j [δi (Nja ) − δj (Nia )]∂˙a − δi (X˙ a )∂˙a − X˙ a ∂˙a (Nib )∂˙b , [X, ∂˙a ] = −∂˙a (X k )δk + X k ∂˙a (Nkb )∂˙b − ∂˙a (X˙ b )∂˙b . It follows from (3.8) that √ √ (divX) g G = (LX dV )(δ1 , · · · , δm , ∂˙1 , · · · , ∂˙r ) = X(dV (δ1 , · · · , δm , ∂˙1 , · · · , ∂˙m )) − −

and we have

m X i=1

r X

a=1

dV (δ1 , · · · , δi−1 , [X, δi ], δi+1 , · · · , δm , ∂˙1 , · · · , ∂˙m ) dV (δ1 , · · · , δm , ∂˙1 , · · · , ∂˙a−1 , [X, ∂˙a ], ∂˙a+1 , · · · , ∂˙r )

n o√ √ √ √ = X( g G) + δi (X i ) − [X k ∂˙a (Nka ) − ∂˙a (X˙ a )] g G

√ √ √ √ divX = [X i δi (ln g)+δi (X i )−X i [∂˙a (Nia )−δi (ln G)]+[X˙ a ∂˙a (ln G)+ ∂˙a (X˙ a )]+ X˙ a ∂˙a (ln g). This completes the proof. Now if we define gradf by G(gradf, X) = Xf, ∀X ∈ X (E),

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then in the adapted frame {δi , ∂˙a }, we have gradf = gradh f + gradv f, where gradh f = g ij (δj f )δi = g ij (∇δj f )δi , grad f = g ab (∂˙b f )∂˙a = g ab (∇ ˙ f )∂˙a . v

(3.12)

∂a

The Laplace operator of a scalar field, f ∈ C ∞ (E), is then defined as △f = div ◦ gradf.

(3.13)

Now the h(v)-Laplace operator on E is defined by △h := divh ◦ gradh

and △v := divv ◦ gradv .

We have Proposition 3.2 Let f ∈ C ∞ (E). Then △f = △h f + △v f,

(3.14)

where 1 √ △h f = √ δi [ gg ij (δj f )] − g ij (δj f )Pi = g ij ∇δi ∇δj f − Pi g ij ∇δj f, g √ 1 △v f = √ ∂˙a [ GG ab (∂˙b f )] + G ab (∂˙b f )P˙a = G ab ∇∂˙a ∇∂˙b f + P˙a G ab ∇∂˙b f. G

(3.15)

Proof It follows from (3.10)–(3.13) directly. We point out here that if the Riemannian structure, G, on E = T M is induced by the Finsler metric F on M , then, by gij = Gab and (3.7), we have √ √ j Pi = ∂˙a (Nia ) − δi (ln G) = Pijj , P˙a = ∂˙a (ln g) = Cij .

(3.16)

Proposition 3.3 If the Riemannian structure G on E = T M is the Sasaki-type metric, which is induced by the Finsler metric F on M , then we have divh C = 0, divv C = n, divh S = divv S = 0, △h E = 0, △v E = 2n. Proof It follows from the proof of Theorem 1.1 in [7] and our definition of h(v)-divergence and h(v)-Laplace operator. The expression of △h f is also obtained in [11–12]. Furthermore, if f ∈ C ∞ (E) or f ∈ ∗ ∞ π C (M ), one obviously obtains that △v f = 0. This motivates us to define the h-harmonicity on the total bundle E: A scalar field f ∈ C ∞ (E) that satisfies △h f = 0 is called a horizontal harmonic scalar field, or briefly, h-harmonic scalar field. Thus the absolute energy of a Finsler space is h-harmonic. It is a main result of [7] that the absolute energy of a Finsler space E cannot be harmonic in the usual sense. i Notice that if (M, F ) is a Landsberg space, we have Pjk = 0, and by (3.7) and (3.11), j Pi = Pji = 0. Thus we have

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Theorem 3.4 If the Riemannian structure, G, on T M comes from a Landsberg space (M, F ), then the h-Laplace operator, △h , has the following form: 1 √ △h f = √ δi [ gg ij (δj f )] = g ij ∇δi ∇δj f, f ∈ C ∞ (E). g Let π : E → M be a real vector bundle with a Riemannian structure, G. Notice that (divX)dV = d[i(X)dV ] for any vector field X ∈ X (E), and since any vector field, X ∈ X (E), can be decomposed into its horizontal and vertical part, we immediately have (divX)dV = d[i(X)dV ], X ∈ XH (E) or X ∈ XV (E). Thus we have Proposition 3.5 In a compact orientable Riemannian vector bundle, E, we have Z (∇δi X i − Pi X i )dV = 0, X = X i δi ∈ XH (E) (3.17) E

and

Z

E

(∇∂˙a Y a − P˙a Y a )dV = 0, Y = Y a ∂˙a ∈ XV (E).

Theorem 3.6 In a compact orientable Riemannian vector bundle, E, we have Z g ij [∇δi ∇δj f − Pi g ij ∇δj f ]dV = 0 E

for any scalar field f on E. Since we have 1 ij g ∇δi ∇δj f 2 − Pi ∇δj f 2 = f g ij ∇δi ∇δj f + g ij (∇δi f )(∇δj f ) − f g ij (∇δi f )Pj , 2 applying Theorem 3.6, we have Theorem 3.7 In a compact orientable Riemannian vector bundle, E, we have Z [f g ij ∇δi ∇δj f + g ij (∇δi f )(∇δj f ) − f g ij (∇δi f )Pj ]dV = 0 E

for any scalar field f on E. If a scalar field f on E satisfies g ij ∇δi ∇δj f − g ij (∇δi f )Pj ≥ 0 in the total space E, then dV has a definite sign. Thus Theorem 3.6 gives g ij ∇δi ∇δj f − g ij (∇δi f )Pj = 0 and consequently Theorem 3.7 gives g ij (∇δi f )(∇δj f ) = 0, from which we have ∇δi f = 0,

(3.18)

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namely, f is horizontally constant, since the form g ij (∇δi f )(∇δj f ) is positive definite. Thus we have Theorem 3.8 If, in a compact orientable Riemannian vector bundle E, we have g ij ∇δi ∇δj f − g ij (∇δi f )Pj ≥ 0 for a scalar field f on E, then f is horizontally constant in the total space E, namely, δi f = 0. If a scalar field f , not horizontally constant on E, satisfies g ij ∇δi ∇δj f − g ij (∇δi f )Pj = cf

(3.19)

with c being a constant and cannot be zero. Substituting this in (3.15), we find Z [cf 2 + g ij (∇δi f )(∇δj f )]dV = 0. E

Thus, if c > 0, then we should have f = 0, and we have Theorem 3.9 If, in a compact orientable Riemannian vector bundle E, a scalar field, f , which is not horizontal constant satisfies g ij ∇δi ∇δj f − g ij (∇δi f )Pj = cf, with c being a constant, then the constant c is negative. Now if we apply the horizontal Laplacian, △h , to the square of the length of a horizontal Finsler vector field, X k (x, y), and notice that ∇δi g jk = 0, we have 1 ij {g ∇δi ∇δj (X k Xk ) − g ij [∇δi (X k Xk )]Pj } 2 = g ij (∇δi ∇δj X k )Xk + g ij (∇δi Xk )(∇δj X k ) − g ij [∇δi X k ]Xk Pj . Thus, applying Theorem 3.6, we have Theorem 3.10 In a compact orientable Riemannian vector bundle, E, we have Z (3.20) {g ij (∇δi ∇δj X k )Xk + g ij (∇δi Xk )(∇δj X k ) − g ij [∇δi X k ]Xk Pj }dV = 0 E

for any horizontal Finsler vector field, X k (x, y). If the second covariant derivative, ∇δi ∇δj X k , of the horizontal Finsler vector, X k , vanishes and Pj = 0, then g ij (∇δi Xk )(∇δj X k ) is positive definite. We have from (3.20) that ∇δi X k = 0, that is, the first horizontal covariant derivative vanishes. Theorem 3.11 If, in a compact orientable Riemannian vector bundle E, the second covariant derivative of a horizontal vector vanishes and Pj = 0, then the first horizontal covariant derivative vanishes too. If a horizontal Finsler vector field X k (x, y), which is not zero, satisfies g ij (∇δi ∇δj X k ) − g ij ∇δi X k ]Pj = cX k

(3.21)

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with c being a constant, substituting (3.21) into (3.20), we have Z [cX k Xk + g ij (∇δi Xk )(∇δj X k )]dV = 0. E

Thus, X k is different from zero and c cannot be positive. Hence, we have Theorem 3.12 If, in a compact orientable Riemannian vector bundle E, a horizontal Finsler vector field, X k (x, y), other than zero vector satisfies g ij (∇δi ∇δj X k ) − g ij ∇δi X k ]Pj = cX k , with c being a constant,then c is negative. Consider a horizontal Finsler vector field, X i (x, y), in a compact orientable Riemannian vector bundle E and calculate the divergence of the vector field X i (∇δi X j ) − X j (∇δi X i ). Then we obtain ∇δj [X i (∇δi X j ) − X j (∇δi X i )] = X j (∇δi ∇δj − ∇δj ∇δi )X i + (∇δj X i )(∇δi X j ) − (∇δi X i )2

= X j {∇[δi ,δj ] + Ω(δi , δj )}X i + (∇δj X i )(∇δi X j ) − (∇δi X i )2 .

Theorem 3.13 In a compact orientable Riemannian vector bundle E, we have Z [X j {∇[δi ,δj ] + Ω(δi , δj )}X i + (∇δj X i )(∇δi X j ) − (∇δi X i )2 ]dV = 0 E

for any horizontal Finsler vector field, X i (x, y).

4

Horizontal Laplace Operator for Finsler Forms on E

In this section, we are concerned with the Laplace operator, △, on horizontal Finsler forms on E. We are mainly interested in the horizontal Laplace operator, △H , for horizontal Finsler forms. We first give a Riemannian inner product in the space of the horizontal Finsler pforms, then we give the local expression of the adjoint operator of dH with respect to this inner product and finally, we define △H for horizontal Finsler forms. At the end of this section, we give the expression of △H in terms of the horizontal covariant derivatives of the metric Finsler connection ∇. Now, let ϕ(x, y) ∈ ∧pH (E) and ψ(x, y) ∈ ∧p+1 H (E). Locally, ϕ=

1 ϕi ···i (x, y)dxi1 ∧ · · · ∧ dxip p! 1 p

Set hϕ, ψi =

and ψ =

1 ψi ···i (x, y)dxi1 ∧ · · · ∧ dxip . p! 1 p

X 1 i1 ···ip ϕ ψi1 ···ip = ϕi1 ···ip ψi1 ···ip , p! i <···
p

where ϕi1 ···ip = ϕj1 ···jp g j1 i1 · · · g jp ip .

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Since ϕ, ψ are horizontal Finsler tensor fields, the above inner product is independent of the local coordinates on E, thus hϕ, ψi is a global inner product on E. Especially, |ϕ|2 = hϕ, ϕi =

1 i1 ···ip ϕ ϕi1 ···ip ≥ 0. p!

Now we define the global inner product Z Z (ϕ, ψ) = hϕ, ψidV and kϕk2 = hϕ, ϕidV. E

(4.1)

E

Consider a covariant tensor of rank p, which has ϕi1 ···ip as components on E, and its horizontal covariant derivative is a new horizontal Finsler covariant tensor of rank p + 1 which has p X (4.2) ∇δi ϕi1 ···ip = δi (ϕi1 ···ip ) − ϕi1 ···iν−1 aiν+1 ···ip Fiaν i ν=1

Fiaν i

as its components, where are the horizontal connection coefficients of the metric Finsler connection. If we denote by k1 · · · b kν · · · kp+1 the sequence got from k1 · · · kp+1 by suppressing kν , we have p+1 X (dH ϕ)k1 ···kp+1 = (−1)ν−1 δkν (ϕk1 ···bkν ···kp+1 ). (4.3) ν=1

i i Noting the symmetry of the horizontal connection coefficients, Fjk = Fkj , we can replace the horizontal derivatives by the horizontal covariant derivatives and obtain

(dH ϕ)k1 ···kp+1 =

p+1 X

(−1)ν−1 ∇δkν ϕk1 ···bkν ···kp+1 .

(4.4)

ν=1

To establish the expression for δH , namely, the codifferential of dH , let ϕ and ψ be two horizontal Finsler forms of rank p and p+ 1, respectively, on E, then by (3.17) and the compactness of E, we have Z h i ∇δi (ϕi1 ···ip ψ ii1 ···ip ) − ϕi1 ···ip ψ ii1 ···ip Pi dV = 0. E

By (3.5), the horizontal covariant derivatives of gij and g ij are zero, and we have Z  Z  ii1 ···ip (∇δi ϕi1 ···ip )ψ dV = − ∇δi ψ ii1 ···ip − ψ ii1 ···ip Pi ϕi1 ···ip dV E E # Z "   i1 j1 ip jp ij =− g ···g g ∇δi ψjj1 ···jp − ψjj1 ···jp Pi ϕi1 ···ip dV. E

Now, the term on the left hand side reduces to p!(dH ϕ, ψ), and by putting (δH ψ)i1 ···ip = −g ki [∇δk ψii1 ···ip − ψii1 ···ip Pk ],

(4.5)

the right-hand side term reduces to p!(ϕ, δH ψ). So we have (dH ϕ, ψ) = (ϕ, δH ψ) for any horizontal Finsler form ϕ ∈ ∧pH (E) and ψ ∈ ∧p+1 H (E). Now, we define △H = dH ◦ δH + δH ◦ dH , ∧pH (E) → ∧pH (E). (4.6)

No.1

Zhong & Zhong: OPERATOR IN REAL FINSLER VECTOR BUNDLES

139

△H is a differential operator, called horizontal Laplace operator for horizontal Finsler forms on E. A horizontal Finsler vector field, ϕ = ϕi dxi ∈ ∧1H (E), satisfying △H ϕ = 0 is called h-harmonic Finsler 1-form on E. Now, we have the following Weitzenb¨ock formula for △H on horizontal Finsler forms on E. Theorem 4.1 The h-Laplace operator, △H , for horizontal Finsler forms on E has the following form: p h i X (△H ϕ)i1 ···ip = −g ik ∇δk ◦ ∇δi ϕi1 ···ip + (−1)ν−1 (∇δiν ∇δk − ∇δk ∇δiν )ϕii1 ···ibν ···ip

+g

ik

h

∇δi ϕi1 ···ip +

p X

ν=1

ν=1

i ∇δiν ϕii1 ···ibν ···ip Pk , ϕ ∈ ∧pH (E).

(4.7)

Proof Since by (4.5), we have

(δH ϕ)i1 ···ip−1 = −g ik [∇δk ϕii1 ···ip−1 − ϕii1 ···ip−1 Pk ], and thus by (4.4), we get (dH ◦ δH ϕ)i1 ···ip = =

p X

(−1)ν g ik ∇δiν (∇δk ϕii1 ···ibν ···ip − ϕii1 ···ibν ···ip Pk )

ν=1 p X

(−1)ν g ik ∇δiν ∇δk ϕii1 ···ibν ···ip −

ν=1

On the other hand, we have (δH ◦ dH ϕ)i1 ···ip = −g ik ∇δk ∇δi ϕi1 ···ip − 

+g ik ∇δi ϕi1 ···ip +

p X

(−1)ν g ik ∇δiν (ϕii1 ···ibν ···ip Pk ).

(4.8)

ν=1

p X

(−1)ν g ik ∇δk ∇δiν ϕii1 ···ibν ···ip

ν=1

p X

 (−1)ν ∇δiν ϕii1 ···ibν ···ip Pk .

(4.9)

ν=1

By (4.8) and (4.9), we complete the proof. In particular, for a form of degree zero, namely, a scalar field f ∈ C ∞ (E), we have dH ◦ δH f = 0 and △H f = δH ◦ dH f = −g ik ∇δk ∇δi f + g ik (∇δi f )Pk .

(4.10)

Thus, we have the similar relationship between h-Laplace operator, △h , for scalar fields and h-Laplace operator, △H , for horizontal Finsler forms on E: △H f = − △h f.

(4.11)

Similarly, if (M, F ) is a Landsberg space, then E = T M with the Sasaki-type metric G i has the property Pjk = 0 and by (3.7) and (3.11), we have Pi = Pjij = 0. In this case we have the following Weitzenb¨ock formula for △H on horizontal Finsler forms on E = T M . Corollary 4.2 If the Riemannian structure G on T M comes from a Landsberg space (M, F ), then the h-Laplace operator, △H , for horizontal Finsler forms on T M has the following form: p h i X ik (△H ϕ)i1 ···ip = −g ∇δk ◦ ∇δi ϕi1 ···ip + (−1)ν−1 (∇δiν ∇δk − ∇δk ∇δiν )ϕii1 ···ibν ···ip . (4.12) ν=1

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From (4.12), we can see that the Weitzenb¨ock formula of horizontal Laplace operator △H for horiontal Finsler forms on E = T M is very similar to that of the usual Laplace operator for forms in Riemannian manifolds [13–14]. Remark 4.1 If E = T M is endowed with a Sasaki metric induced by the given Riemannian metric on M , then the non-linear connection on the bundle E coincides with the Levi-Civita connection on M . If, furthermore, we restrict ϕ to be the pull-back of forms on M , then the horizontal covariant derivative in this case coincide with the usual covariant derivative of the Levi-Civita connection. Thus, one can look Theorem 4.1 and Corollary 4.2 as the natural generalization of the Laplace operator from Riemannian manifold [13–14] to Finsler vector bundles. References 1 Aikou T. Differential geometry of Finsler vector bundles. Rep Fac Sci Kagoshima Univ, (Math, Phys & Chem), 1992, 25: 1–20 2 Bochner S. Vector fields and Ricci curvature. Bull Amer Math Soc, 1946, 52: 776–797 3 Wu H. The Bochner technique in differential geometry. Advance in Mathematics (Chinese), 1981, 10(1): 57–76 4 Miron R. Techniques of Finsler geometry in the theory of vector bundles. Acta Sci Math, 1985, 49: 119–129 5 Abate M, Patrizio G. Finsler Metrics—A Global Approach with Applications to Geometric Function Theory. Lecture Notes in Mathematics, Vol 1591. Berlin: Springer-Verlag, Berlin Heidelberg, 1994 6 Bao D, Chern S S, Shen Z. An introduction to Riemann-Finsler geometry. New York: Springer-Verlag, New York Inc, 2000 7 Anastasiei M, Kawaguchi H. Absolute energy of a Finsler space. Tensor, N S, 1993, 53 8 Sasaki S. On the differential geometry of tangent bundle of Riemannian manifolds. Tˆ ohoku Math J, 1958, 10: 338–354 9 Aikou T. Some remarks on the geometry of tangent bundle of Finsler manifolds. Tensor N S, 1993, 52 10 Ichijy¨ o Y, Lee I -Y, Park H -S. On generalized Finsler structures with a vanishing hv-torsion. J Korean Maht Soc, 2004, 41(2): 369–378 11 Bao D, Lackey B. A geometric inequality and a Weitzenb¨ ock formula for Finsler surfaces. In: The Theory of Finslerian Laplacians and Applications, MAIA 459. Kluwer Academic Publishers, 1998. 245–275 12 Munteanu O. Weitzenb¨ ock formulas for horizontal and vertical Laplacians. Houston Journal of Mathematics, 2003, 29(4): 889-900 13 De Rham G. Vari´ et´s Diff´ erentielles. Paris: Hermann, 1955 14 Yano K. Differential geometry on complex and almost complex spaces. Pergamon Press Limited, 1965