On Berwald Spaces which Satisfy the Relation Γ k ij = p k g ij for Some Functions p k on TM

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1 International Mathematical Forum, 2, 2007, no. 67, On Berwald Spaces which Satisfy the Relation Γ k ij = p k g ij for Some Functions p k on TM Dariush Latifi and Asadollah Razavi Faculty of Mathematics and Computer Science Amirkabir University of Technology 424 Hafez Ave., Tehran, Iran dlatifi@aut.ac.ir, arazavi@aut.ac.ir Abstract Let A ijk and Rjkl a denote the Cartan tenr and curvature tenr respectively. In this paper we show that any Berwald space which satisfies the relations Γ k ij = pk g ij for me function p k and A iju Rkl u =0is a Riemannian space of constant sectional curvature. In particular we give an explicit formula for these metrics. Keywords: Berwald space; Christoffel symbols; Locally Minkowski space; Sectional curvature 1 Introduction Finsler geometry is a generalization of Riemannian geometry. In Finsler geometry, distance ds between two neighboring points x k and x k + dx k,k =1,..., n is defined by a scale function ds = F (x 1,..., x n ; dx 1,..., dx n ) or simply ds = F (x k,dx k ), k =1,..., n. The Riemannian space with metric tenr g ij (x k ) is nothing but the Finsler space with scale function F (x k,dx k )= [ g ij (x k )dx i dx j] 1 2. Among the Finsler spaces, Berwald spaces form a very important class. A Finsler space is called Berwaldian if the Chern connection defines a linear connection directly on the underlying manifold [2]. Berwald spaces are only a bit more general than Riemannian spaces and locally Minkowskian spaces. Since the connection is linear, its tangent spaces are linearly imetric to a common

2 3332 D. Latifi and A. Razavi Minkowski space. Therefore Berwald spaces behave very much like Riemannian spaces. Since Riemannian spaces have many applications in certain area of physics, it is hopeful that Berwald metrics will be useful in the study of me physical problems [1]. In this paper we consider Berwald spaces which in a coordinate system x i, the Christoffel symbols satisfies the relation Γ k ij = p k g ij. 2 Preliminaries We will use the following notations. M is a manifold of dimension n with local coordinate (x i ),i =1,..., n around a point x. We al use the Einstein s convention for summations. If X T x M, then X = X i, where the X i are x i coordinates for the tangent bundle canonically induced from the x i for the base manifold. F is a Finsler metric, i.e. a function F : TM R where 1. F is positive-definite : F (x, X) 0, with equality iff X = o 2. F is smooth except on the zero-section: F TM\{(x, o ) x M} is C. 3. F is strictly convex: at any (x, X), rank [ 2 F X i X j ] = n F is positively homogeneous : F (x, λx) =λf (x, X) for all λ>0. Finsler metrics are a natural generalization of Riemannian metrics. Finsler space (M,F) is Riemannian if and only if F has the form A F 2 (x, X) =g ij (x)x i X j, where the coefficients g ij are independent of the tangent vector X. When this happens, every unit ball, at any point x, I x = {X T x M F (X) 1} is an ellipid. Several commonly used Finsler quantities appear al in Riemannian spaces, and generally have the same interpretation there, though most Finsler quantities are functions of TM rather than M. Some frequently used quantities and relations are: g ij (x, X) := 1 2 F 2 (x, X) 2 X i X j F 2 (X i x )=g ij(x, X)X i X j i

3 Berwald spaces 3333 g jk g ij = δ k i Γ i jk(x, X) := 1 ( gjr 2 gir (x, X) x (x, X) g ) jk k x (x, X)+ g rk (x, X) r xj To simplify the computation, we introduce the adapted bases for the bundles T (TM \{o}) and T (TM \{o}) as: where {dx i, δyi F = 1 F (dyi + N i mdx m )}, { δ δx i = Nm i = 1 ( ( g is gsk 4 y m x j x i N m i + g sj x g ) ) kj y j y k k x s y,f m y } i In fact they are dual to each other. The vector space spanned by δ (resp. F ) δx i y i is called horizontal(resp. vertical) subspace of T (T M \{o}). There are several notions of curvature in Finsler geometry. The notion of Riemannian curvature in Riemannian geometry can be extended to Finsler metrics. The Riemannian curvature tenr or hh-curvature tenr with components Rjkl i is defined by Rjkl i = δγi jl δx δγi jk k δx l R i jkl satisfies the following properties R i jkl = Ri jlk, +Γ i hkγ h jl Γ i hlγ h jk Rjkl i + Rklj i + Rljk i =0. We have the following relation R ijkl + R jikl =2( A iju R u kl) where A iju is the component of Cartan tenr and denote A iju R u kl by B ijkl. Definition 2.1 Let (M,F) be a Finsler space. space if, for any x M, and y T x M, y l ( Γ i jk ) (x,y) =0; Then (M,F) is a Berwald equivalently, we could say that the Christoffel symbols Γ i jk base point x and not on the tangent vector y. depend only on the

4 3334 D. Latifi and A. Razavi 3 Main results Let (M,F) be an n-dimensional Berwald space. Suppose that there is a coordinate system (x i ) such that the Christoffel symbols satisfy the relation Γ k ij = p k g ij for me functions p k on TM. If we set p i = g ki p k and g = det(g ij ), then the relation Γ i ik = δ (log g) implies that p k = that p k s are components of a gradient. In Finsler space we have the following Ricci identity λ i jk λ i kj = λ r R r ijk Rt kj F λ i y t δ (log g), this shows where λ i j is the horizontal covariant derivative of λ i with respect to δ.now δx j since we have p i = g ki p k =Γ k ki and Γk ki just depend on x, p i = 0 and Ricci y t identity reduce to p i jk p i kj = p r Rijk r. Now writing B for 1 ( δpa ). In the following lemma we prove that B is n δx a constant along all horizontal curve. Lemma 3.1 If (M,F) is a Berwald space satisfying the relations Γ k ij = pk g ij and B jikl =0then, =0 Proof: From the relation Γ k ij = pk g ij we compute the Riemannian curvature tenr of (M,F) Rjkl a = g jl δx g k jk δx l R jikl = g ia Rjkl a = g ia g jl from the relation R jikl = R ijlk we have g iag jk δx l, g ia g jl δx g iag k jk = g δx l ja g ik δx g jag l il By multiplication by g jl we have : then g jl g ia g jl δx k gji g ia g jk = g jl g δx l ja g ik δx l gjl g ja g il ng ia δx g iaδ l k k = g δx l ik δa l δx l δl a g il

5 Berwald spaces 3335 thus Multiplication by g ih gives : ng ia δx g k ia δx = g k ik δx g a ia g ik B = g ia g ih g ia δx = k gih g ik B δp h δx = k δh k B. With the aid of the covariant derivative we get further covariant differentiation yields p a j = δpa δx j + p jp a p i j = g ai p a j = g ij B + p i p j p i jk = g ij + δp i p j + p i δp j p rp j p r g ik p i p s p s g jk p i jk p i kj = g ij + δp i p j+ δp j p i p r p j p r g ik g ik δx j δp i δx j p k δp k δx j p i+p r p k p r g ij To simplify the above equality we compute δp i : δp i = δg ir pr + g ir δp r = ( g nr Γ n ik + g itγ t rk = p r p r g ik + p k p i + g ik B ) p r + g ir δp r Thus p i jk p i kj = g ij δx g k ik δx + g ikp j j B g ij p k B (1) Now from the Ricci identity p i jk p i kj = p a R a ijk (2) and Rijk a = g ik δx g j ij

6 3336 D. Latifi and A. Razavi we have So from (1) and (2) we have then we obtain p a R a ijk = p ag ik δ a j B p ag ij δ a k B = p j g ik B p k g ij B p j g ik B p k g ij B = g ij δx g k ik δx + g ikp j j B g ij p k B g ij δx = g k ik δx j g ij g ij = gij g ik δx j n = Thus = 0 Q.E.D. Theorem 3.1 If in a n-dimensional Berwald space there is a coordinate system such that the Christoffel symbols satisfy the relation Γ k ij = pk g ij,b jikl =0 for me functions p k on TM, then δ δx i Ra jkl =0, where is the covariant derivative defined by Γ k ij. Proof: Since the space is of Berwald type it is enough to show that δ δx i Ra jk = 0 [4]. From definition we have Now since that R a jk = l i R a ijk = l i g ik δx j li g ij = l i g ik δ a j B li g ij δ a k B = l k δ a j B l j δ a kb = 0 and δ l i dx i = 0 with a direct computation we conclude δ δx i Ra jk =0 Q.E.D.

7 Berwald spaces 3337 Corollary 3.1 A Berwald space admitting the condition of theorem 3.1 is projectively symmetric space that is if Wjkh i be the projective curvature tenr of this space then δ W i δx l jkh =0 Proof: Following the theorem 3.1 this space satisfies δ R i δx l jkh = 0 from [5] we have δ W i δx l jkh = 0 Q.E.D. Theorem 3.2 If for an n-dimensional Berwald space the conditions Γ k ij = p k g ij and B jikl =0, hold in a coordinate system, then this space has constant flag curvature B. Proof: We have R jikl = g ia g jl g iag jk δx l δp and with the help of the relation g a ia = g ik B and δph that R jikl = g jl g ik B g jk g il B = δ h kb we conclude R ik = l j R jikl l l = l l l l g ik B l k l i B = (g ik l k l i )B g ik l k l i is called angular metric and shown by h ik thus R ik = h ik B. From the theorem of [2] this space has scalar flag curvature. A result given in [1] says that if M is connected and F is of scalar flag curvature λ(x, y) and λ i = 0 then λ must in fact be constant. So from the relation =0,we δx i conclude that B must be constant. Q.E.D. In the previous theorem, we have investigated me geometric properties of Berwald space with the property Γ k ij = pk g ij. Now we will give me partial answers about existence of this spaces. If (M,F) be a locally Minkowski space then at every point x M, there is a local coordinate system (x i ), with induced tangent space coordinate y i, such that F has no dependence on the x i. In this coordinates the Christoffel symbols Γ i jk vanish identically. So we can choose pi identically zero i.e p i =0, and evidently B jikl = 0. Riemannian space with property Γ k ij = pk g ij have been detected in [3].

8 3338 D. Latifi and A. Razavi Theorem 3.3 If for a n-dimensional Berwald space the conditions Γ k ij = pk g ij and B jikl =0hold in a coordinate system, Then this space is Riemannian space with constant sectional curvature. Proof: In the theorem 3.2 we showed that B is constant. So From the relation δpa = δ a k B we conclude that pi = Bx i is a answer for it. This tell us that we can choose p i s with no dependence on the y i. Hence the original relations Γ k ij = pk g ij and B jikl = 0 imply that g ij has no dependence on the y i, thus this space is Riemannian. Q.E.D. Remark 3.1 Now we determine the Riemannian metric tenr g ij satisfying the relation Γ k ij = pk g ij. x j gik =0implies that g ik x j whence it follows that = δ k j pi δ i j pk = Bx i δ k j Bxk δ i j g ik = Bx i x k + c ik The c s being arbitrary symmetric constant, we fined References g ij = c ij + Bc irx r c js x s (1 Bc ab x a x b ) [1] P.L. Antonelli, R.S. Ingarden and M. Matsumoto, The theory of sprays and Finsler spaces with Applications in Physics and Biology. FTPH 58, Kluwer Academic Publishers [2] D. Bao,S.-S. Chern and Shen,An Introduction to Riemann-Finsler geometry. Springer-Verlag,New-York.2000 [3] C.M. Fulton, A theorem on spaces of constant curvature. Amer.Math.Montly. 74: (1967) [4] D. Latifi, A. Razavi, A Symmetric Finsler Space With Chern Connection. Proceedings of the 3rd Seminar on Geometry and Topology, Azerb. Univ. Tarbiat Moallem,Tabriz (2004) [5] R.B. Misra, A projectively symmetric Finsler space. Math.Z. 126: (1972) Received: June 16, 2007