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
Lx,ky=k2Lx,y ∀k >0. (2.1) (L3) The metric tensor
gijx,y=12 ∂
2L
∂yi∂yj (2.2)
hasqnegative eigenvalues andm-qpositive eigenvalues for all(x,y)∈TM. Then we say thatFm=(M,L)is apseudo-Finsler manifoldof indexq. If, in particular,
q=0,Fmbecomes a Finsler manifold (cf. Rund [9, p. 5]).
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 byT0
2(VTM)the vector bundle overTMof all bilinear mappings onVTM. Then, from (2.2),{g
ij}define a global section ofT20(VTM)given onUby g(X,Y )x,y=gijx,yXix,yXjx,y ∀X,Y∈ΓVTM. (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)=gijx,yyiyj. (2.6)
Thus, taking account of (2.4) and (2.5) in (2.6), we obtainL >0 onTM. The funda-mental functionofFm (cf. Matsumoto [7, P. 101]) is defined byF=L1/2, 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∂yj=0. (2.8)
Moreover, sinceL=F2, from (2.6), we get gij=F ∂
2F ∂yi∂yj+
∂F ∂yi
∂F
∂yj. (2.9)
By contracting (2.9) byyjand taking account of (2.7) and (2.8), we obtain
gijyj=F∂y∂Fi. (2.10)
Similarly, sinceLis positive homogeneous of degree two with respect toy, we have
yi ∂L
which implies that
yi ∂2L
∂yi∂yj=
∂L
∂yj. (2.12)
Finally, differentiating (2.12) with respect toyk, we obtain
yi ∂3L
∂yi∂yj∂yk=0, (2.13)
which yields
∂gij
∂ykyi=0,
∂gij
∂ykyj=0,
∂gij
∂ykyk=0. (2.14)
Next, we defineξ=(1/F)V and by using (2.3), (2.4), and (2.6), we get
gξ,ξ=1. (2.15)
By means ofgandξ, we define the Finsler 1-formηby
η(X)=gX,ξ ∀X∈ΓVTM. (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
ΓSTM= {X∈ΓVTM; 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∈ΓVTM. (2.19) Denote by{wi}the dual basis inΓV∗TM|
Uwith respect to{∂/∂yi}. Then the local components ofηandPwith respect to the basis{wi}and{wi⊗∂/∂yj}, respectively,
are given by
ηi=∂y∂Fi (2.20)
and
Pij=δji−1Fηiyj, (2.21)
whereδji are the components of the Kronecker delta.
TheindicatrixofFmis 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 Fm via the Liouville distribution.
In this section, we suppose thatFm is a pseudo-Finsler manifold which is never a
semi-Riemannian manifold, that is{gij}does not depend on(xi)alone.
Theorem3.1. LetFm=(M,L)be a pseudo-Finsler manifold. Then the Liouville
dis-tribution ofFmis 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
gijx,yyiXjx,y=0, (3.1)
whereXjare the components ofX. Differentiate (3.1) with respect toykand use (2.14)
to obtain
gkjx,yXjx,y+gijx,yyi∂X j
∂yk
x,y=0. (3.2)
Then by direct calculations using (2.3), (2.4), and (3.2), we infer that
g[X,Y ],ξ=F1gijyi
∂Yj
∂ykXk−
∂Xj
∂ykYk
=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),g
ij(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 Ck
ijof∇. By using (2.2) and the usual formula forCijk (see O’Neill [8, P. 62]), we obtain
Ck ij
a,y=12gkha,y∂ghi
∂yj
a,y, (3.4)
where{gkh(a,y)}are the entries of the inverse matrix of them×mmatrix[g
kh(a,y)].
Contracting (3.4) byyj, we deduce that
Ck ij
a,yyj=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. LetFm=(M,L)be a pseudo-Finsler manifold. Then,on any leafNof VTM,we have
∇Xξ=1FPX, (3.6)
∇XηY =F1g(PX,PY ), (3.7)
and
(∇XP)Y= −F1g(PX,PY )ξ+η(Y )PX (3.8)
for anyX,Y∈Γ(TN).
Theorem3.2. LetFm=(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 ofFmand 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= −F1g(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 =gij ∂F
∂yj (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
εiBEi,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. LetFm=(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
RX,ξξ= 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. LetFm=(M,L)be a pseudo-Finsler manifold. Then there exist no
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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
