ON THE VERTICAL BUNDLE OF A PSEUDO-FINSLER MANIFOLD

Vol. 22, No. 3 (1999) 637–642 S 0161-17129922637-3

©Electronic Publishing House

ON THE VERTICAL BUNDLE OF A PSEUDO-FINSLER MANIFOLD

AUREL BEJANCU and HANI REDA FARRAN

(Received 21 August 1996 and in revised form 24 November 1997)

Abstract.We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.

Keywords and phrases. Pseudo-Finsler manifold, Liouville distribution, vertical bundle.

1991 Mathematics Subject Classification. 53B50, 53B30.

1. Introduction. As it is well known, the vertical vector bundleVTM of a manifold M is an integrable distribution on the tangent bundleTM ofM. The present paper is concerned with the study of the geometry of leaves ofVTM in the case when the base manifoldMcarries a pseudo-Finsler structure. In our study, the pseudo-Finsler metric is, in fact, considered as a semi-Riemannian metric onVTM. This enables us to define the Liouville distribution which will be the important tool in studying the leaves ofVTM.

The main results are stated in Theorems 3.2 and 3.3.

2. Preliminaries. LetM be a smooth (C⁴is enough)m-dimensional manifold and TM be the tangent bundle ofM. Denote byΘthe zero section ofTM and setTM= TM\Θ(M). The coordinates of a point ofTMare denoted by(xⁱ,yⁱ), where(xⁱ)and (yⁱ)are the coordinates of a pointx∈Mand the components of a vectoryinTxM, respectively. Consider a continuous functionL(x,y)defined onTMand suppose that the following conditions are satisfied.

(L1) Lis smooth onTM.

(L2) Lis positive homogeneous of degree two with respect toy, i.e., we have L

x,ky

=k²L x,y

∀k >0. (2.1) (L3) The metric tensor

gij x,y

=1 2

∂²L

∂yⁱ∂y^j (2.2)

hasqnegative eigenvalues andm-qpositive eigenvalues for all(x,y)∈TM. Then we say thatF^m=(M,L)is apseudo-Finsler manifoldof indexq. If, in particular, q=0,F^mbecomes a Finsler manifold (cf. Rund [9, p. 5]).

Denote byVTMthe vertical vector bundle overTM, that is,VTM=kerdπ, where π:TM→Mis the canonical projection anddπ is its differential mapping. Then any section ofVTM is aFinsler vector field. Also, any section of the dual vector bundle

V^∗TMis aFinsler 1-form. In this way, the entire Finsler tensor calculus can be devel- oped via the vertical vector bundle (cf. Bejancu [6]).

Denote byF(TM) the algebra of smooth functions on TM and by Γ(VTM)the F(TM)-module of smooth sections ofVTM. We keep the same notation for any other vector bundle. We also use the Einstein convention, that is repeated indices with one upper index and one lower index denote summation over their range.

LetU be a coordinate neighborhood ofTM and X∈Γ(VTM_|U). AsVTM is an integrable distribution onTM, it follows that{∂/∂yⁱ}is a basis ofΓ(VTM_|U), and thusX=Xⁱ(x,y)(∂/∂yⁱ). An important Finsler vector field is defined by

V=yⁱ ∂

∂yⁱ, (2.3)

and it is called theLiouville vector field.

Next, we denote byT₂⁰(VTM)the vector bundle overTMof all bilinear mappings onVTM. Then, from (2.2),{gij}define a global section ofT₂⁰(VTM)given onUby

g(X,Y ) x,y

=gij x,y

Xⁱ x,y

X^j x,y

∀X,Y∈Γ VTM

. (2.4)

Asg is symmetric, condition (L3) enables us to claim thatg is asemi-Riemannian metricof indexqonVTM(cf. O’Neill [8]). Throughout the paper, we suppose thatV is space-like with respect tog, i.e., we have

g(V,V) >0. (2.5)

By using the homogeneity ofL, we deduce that L(x,y)=gij

x,y

yⁱy^j. (2.6)

Thus, taking account of (2.4) and (2.5) in (2.6), we obtainL >0 onTM. Thefunda- mental functionofF^m (cf. Matsumoto [7, P. 101]) is defined byF=L^1/2, and thus it is positive homogeneous of degree one with respect toy. Hence, we have (cf. Bao- Chern [1])

yⁱ ∂F

∂yⁱ=F (2.7)

and

yⁱ ∂²F

∂yⁱ∂y^j=0. (2.8)

Moreover, sinceL=F², from (2.6), we get gij=F ∂²F

∂yⁱ∂y^j+ ∂F

∂yⁱ

∂F

∂y^j. (2.9)

By contracting (2.9) byy^jand taking account of (2.7) and (2.8), we obtain gijy^j=F ∂F

∂yⁱ. (2.10)

Similarly, sinceLis positive homogeneous of degree two with respect toy, we have yⁱ ∂L

∂yⁱ=2L, (2.11)

which implies that

yⁱ ∂²L

∂yⁱ∂y^j= ∂L

∂y^j. (2.12)

Finally, differentiating (2.12) with respect toy^k, we obtain yⁱ ∂³L

∂yⁱ∂y^j∂y^k=0, (2.13)

which yields

∂gij

∂y^kyⁱ=0, ∂gij

∂y^ky^j=0, ∂gij

∂y^ky^k=0. (2.14)

Next, we defineξ=(1/F)V and by using (2.3), (2.4), and (2.6), we get g

ξ,ξ

=1. (2.15)

By means ofgandξ, we define the Finsler 1-formηby η(X)=g

X,ξ

∀X∈Γ VTM

. (2.16)

Denote by{ξ}the line vector bundle overTMspanned byξand define theLiouville distributionas the complementary orthogonal distributionSTMto{ξ}inVTMwith respect tog. Hence,STMis defined byη, that is we have

Γ STM

= {X∈Γ VTM

; η(X)=0}. (2.17)

Thus, any Finsler vector fieldXcan be expressed as follows:

X=PX+η(X)ξ, (2.18)

wherePis the projection morphism ofVTMonSTM. By direct calculations, we obtain g(X,PY )=g(PX,PY )=g(X,Y )−η(X)η(Y ) ∀X,Y∈Γ

VTM

. (2.19)

Denote by{wⁱ}the dual basis inΓ

V^∗TM|U

with respect to{∂/∂yⁱ}. Then the local components ofηandPwith respect to the basis{wⁱ}and{wⁱ⊗∂/∂y^j}, respectively, are given by

ηi= ∂F

∂yⁱ (2.20)

and

P_i^j=δ^j_i−1

Fηiy^j, (2.21)

whereδ^j_i are the components of the Kronecker delta.

TheindicatrixofF^mis the hypersurfaceIofTMgiven by the equationF(x,y)=1.

More about pseudo (indefinite)-Finsler manifolds can be found in a series of papers by Beem [2, 3, 4, 5].

3. Geometry of the vertical vector bundle of F^m via the Liouville distribution.

In this section, we suppose thatF^m is a pseudo-Finsler manifold which is never a semi-Riemannian manifold, that is{gij}does not depend on(xⁱ)alone.

First, we prove the following result.

Theorem3.1. LetF^m=(M,L)be a pseudo-Finsler manifold. Then the Liouville dis- tribution ofF^mis integrable.

Proof. LetX,Y ∈Γ(STM). AsVTMis an integrable distribution onTM,[X,Y ] lies in Γ(VTM). Hence, we need only to show that [X,Y ]has no component with respect toξ. By using (2.4) and (2.16), we deduce thatX∈Γ(STM)if and only if

gij x,y

yⁱX^j x,y

=0, (3.1)

whereX^jare the components ofX. Differentiate (3.1) with respect toy^kand use (2.14) to obtain

gkj x,y

X^j x,y

+gij x,y

yⁱ∂X^j

∂y^k x,y

=0. (3.2)

Then by direct calculations using (2.3), (2.4), and (3.2), we infer that g

[X,Y ],ξ

= 1 Fgijyⁱ

∂Y^j

∂y^kX^k−∂X^j

∂y^kY^k

=0 (3.3)

which completes the proof.

Based on the above results, we may say that the geometry of the leaves ofVTM should be derived from the geometry of the leaves of STM and of integral curves ofξ. In order to get this interplay, we consider a leafN ofVTM given locally by xⁱ=aⁱ,i∈ {1,...,m}, where the aⁱ’s are constants. Then, from (2.2),gij(a,y)are the components of a semi-Riemannian metricgof index qonN. Denote by∇the Levi-Civita connection onNwith respect togand consider the Christoffel symbols C_ij^k of∇. By using (2.2) and the usual formula forC_ij^k (see O’Neill [8, P. 62]), we obtain

C_ij^k a,y

=1 2g^kh

a,y∂ghi

∂y^j a,y

, (3.4)

where{g^kh(a,y)}are the entries of the inverse matrix of them×mmatrix[gkh(a,y)].

Contracting (3.4) byy^j, we deduce that C_ij^k

a,y

y^j=0. (3.5)

By straightforward calculations using (3.5), (2.3), (2.19), (2.20), and (2.21), we obtain the covariant derivatives ofξ,η, andPin the following lemma:

Lemma3.1. LetF^m=(M,L)be a pseudo-Finsler manifold. Then,on any leafNof VTM,we have

∇Xξ=1

FPX, (3.6)

∇Xη Y =1

Fg(PX,PY ), (3.7)

and

(∇XP)Y= −1 F

g(PX,PY )ξ+η(Y )PX

(3.8) for anyX,Y∈Γ(TN).

Now, we state the main results of the paper.

Theorem3.2. LetF^m=(M,L)be a pseudo-Finsler manifold andN,N,andC be a leaf of VTM,a leaf of STMthat lies inN,and an integral curve ofξ,respectively.

Then we have the following assertions. (i) Cis a geodesic ofNwith respect to∇.

(ii) Nis totally umbilical immersed inN.

(iii) Nlies in the indicatrix ofF^mand has constant mean curvature equal to−1.

Proof. ReplaceXbyξin (3.6) and obtain (i). Taking into account thatξis the unit normal vector field ofN, the second fundamental formB ofNas a hypersurface of Nis given by

B(X,Y )=g

∇XY ,ξ

∀X,Y∈Γ(TN). (3.9)

On the other hand, by using (3.6) and taking into account thatgis parallel with respect to∇, we deduce that

g

∇XY ,ξ

= −1

Fg(X,Y ) ∀X,Y∈Γ(TN). (3.10) Hence,

B(X,Y )= −1

Fg(X,Y ), (3.11)

that is,Nis totally umbilical immersed inN. Now, from (2.10), it follows that yⁱ

F =g^ij ∂F

∂y^j (3.12)

which proves thatξis a unit normal vector field for both Nand the componentIa

of the indicatrixIata∈M. Thus,N lies inIa and F(a,y)=1 at any pointy∈N.

Then (3.11) becomes

B(X,Y )= −g(X,Y ) (3.13)

which implies that

1 m−1

m−1

i=1

εiB Ei,Ei

= −1, (3.14)

where{Ei}is an orthonormal frame field onN of signature{εi}. Hence, the mean curvature ofNis−1. The proof is complete.

Theorem3.3. LetF^m=(M,L)be a pseudo-Finsler manifold andNbe a leaf of the vertical vector bundle VTM. Then the sectional curvature of any nondegenerate plane section onNcontaining the Liouville vector field is equal to zero.

Proof. Denote byRthe curvature tensor field of∇onN. Then, by using (3.6) and (3.8), we obtain

R X,ξ

ξ= 1 F²

1−ξ(F)

PX (3.15)

for any unit vector fieldXonN. But from (2.3) and (2.7), we deduce thatξ(F)=1.

Hence, the sectional curvature of a plane section{X,ξ}vanishes onN.

Corrolary3.1. LetF^m=(M,L)be a pseudo-Finsler manifold. Then there exist no leaves of VTMwhich are positively or negatively curved.

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Bejancu: Department of Mathematics, Technical University of Iasi, C. P.17, Iasi1, 6600, Iasi, Romania

Current address:Department of Mathematics and Computer Sciences, Kuwait Univer- sity, P.O. Box5969, Safat13060, Kuwait

E-mail address:[email protected] and [email protected] Farran: Department of Mathematics and Computer Sciences, Kuwait University, P.O.

Box5969, Safat13060, Kuwait

E-mail address:[email protected]