Symmetry, Integrability and Geometry: Methods and Applications On Special Berwald Metrics SIGMA 6 (2010), 008, 9 pages

In this paper, we study a class of Finsler metrics which contains the class of Berwald metrics as a special case. We prove that every Finsler metric in this class is a generalized Douglas-Weyl metric. Then we study isotropic flag curvature Finsler metrics in this class. Finally we show that on this class of Finsler metrics, the notion of Landsberg and weakly Landsberg curvature are equivalent.


Introduction
For a Finsler metric F = F (x, y), its geodesics curves are characterized by the system of differential equationsc i + 2G i (ċ) = 0, where the local functions G i = G i (x, y) are called the spray coefficients. A Finsler metric F is called a Berwald metric if G i = 1 2 Γ i jk (x)y j y k is quadratic in y ∈ T x M for any x ∈ M . It is proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski functionals [7]. Thus Berwald spaces can be viewed as Finsler spaces modeled on a single Minkowski space.
Recently by using the structure of Funk metric, Chen-Shen introduce the notion of isotropic Berwald metrics [6,16]. This motivates us to study special forms of Berwald metrics.
Let (M, F ) be a two-dimensional Finsler manifold. We refer to the Berwald's frame (ℓ i , m i ) where ℓ i = y i /F (y), m i is the unit vector with ℓ i m i = 0, ℓ i = g ij ℓ i and g ij is the fundamental tensor of Finsler metric F . Then the Berwald curvature is given by where I is 0-homogeneous function called the main scalar of Finsler metric and I 2 = I ,2 + I ,1|2 (see [2, page 689]). By (1), we have where h ij := m i m j is called the angular metric. Using the special form of Berwald curvature for Finsler surfaces, we define a new class of Finsler metrics on n-dimensional Finsler manifolds which their Berwald curvature satisfy in following where µ i = µ i (x, y) and λ = λ(x, y) are homogeneous functions of degrees −2 and −1 with respect to y, respectively. By definition of Berwald curvature, the function µ i satisfies µ i y i =0 [12].
The Douglas tensor is another non-Riemanian curvature defined as follows (3) Douglas curvature is a non-Riemannian projective invariant constructed from the Berwald curvature. The notion of Douglas curvature was proposed by Bácsó and Matsumoto as a generalization of Berwald curvature [4]. We show that a Finsler metric satisfies (2) with vanishing Douglas tensor is a Randers metric (see Proposition 1). A Finsler metric is called a generalized Douglas-Weyl (GDW) metric if the Douglas tensor satisfy in h i α D α jkl|m y m = 0 [10]. In [5], Bácsó-Papp show that this class of Finsler metrics is closed under projective transformation. We prove that a Finsler metric satisfies (2) is a GDW-metric. Theorem 1. Every Finsler metric satisfying (2) is a GDW-metric.
Theorem 1, shows that every two-dimensional Finsler metric is a generalized Douglas-Weyl metric.
For a Finsler manifold (M, F ), the flag curvature is a function K(P, y) of tangent planes P ⊂ T x M and directions y ∈ P . F is said to be of isotropic flag curvature if K(P, y) = K(x) and constant flag curvature if K(P, y) = const.
Theorem 2. Let F be a Finsler metric of non-zero isotropic flag curvature K = K(x) on a manifold M . Suppose that F satisfies (2). Then F is a Riemannian metric if and only if µ i is constant along geodesics.
Beside the Berwald curvature, there are several important Finslerian curvature. Let (M, F ) be a Finsler manifold. The second derivatives of 1 2 F 2 x at y ∈ T x M 0 is an inner product g y on T x M . The third order derivatives of 1 2 F 2 x at y ∈ T x M 0 is a symmetric trilinear forms C y on T x M . We call g y and C y the fundamental form and the Cartan torsion, respectively. The rate of change of the Cartan torsion along geodesics is the Landsberg curvature L y on T x M for any y ∈ T x M 0 . Set J y := n i=1 L y (e i , e i , ·), where {e i } is an orthonormal basis for (T x M, g y ).
J y is called the mean Landsberg curvature. F is said to be Landsbergian if L = 0, and weakly Landsbergian if J = 0 [13,14].
In this paper, we prove that on Finsler manifolds satisfies (2), the notions of Landsberg and weakly Landsberg metric are equivalent. There are many connections in Finsler geometry [15]. In this paper, we use the Berwald connection and the h-and v-covariant derivatives of a Finsler tensor field are denoted by "|" and "," respectively.

Preliminaries
Let M be a n-dimensional C ∞ manifold. Denote by T x M the tangent space at Let x ∈ M and F x := F | TxM . To measure the non-Euclidean feature of F x , define C y : The family C := {C y } y∈TM 0 is called the Cartan torsion. It is well known that C = 0 if and only if F is Riemannian [14]. For y ∈ T x M 0 , define mean Cartan torsion I y by I y (u) := I i (y)u i , where I i := g jk C ijk , g jk is the inverse of g jk and u = u i ∂ ∂x i | x . By Deicke's theorem, F is Riemannian if and only if I y = 0 [13].
Let α = a ij (x)y i y j be a Riemannian metric, and Let (M, F ) be a Finsler manifold. Then for a non-zero vector y ∈ T x M 0 , define the Matsumoto torsion M y : This quantity is introduced by Matsumoto [8]. Matsumoto proves that every Randers metric satisfies that M y = 0. Later on, Matsumoto-Hōjō proves that the converse is true too.

Lemma 1 ([9]). A Finsler metric F on a manifold of dimension n ≥ 3 is a Randers metric if and only if
Let us consider the pull-back tangent bundle π * T M over T M 0 defined by Let ∇ be the Berwald connection. Let {e i } n i=1 be a local orthonormal (with respect to g) frame field for the pulled-back bundle π * T M such that e n = ℓ, where ℓ is the canonical section of π * T M defined by ℓ y = y/F (y). Let {ω i } n i=1 be its dual co-frame field. Put is a local basis for T * (T M 0 ). Since {Ω j i } are 2-forms on T M 0 , they can be expanded as The objects R and B are called, respectively, the hh-and hv-curvature tensors of the Berwald connection with the components R(ē k ,ē l )e i = R j ikl e j and P (ē k ,ė l )e i = P j ikl e j [15]. With the Berwald connection, we define covariant derivatives of quantities on T M 0 in the usual way. For example, for a scalar function f , we define f |i and f ·i by where "|" and "," denote the h-and v-covariant derivatives, respectively.
The horizontal covariant derivatives of C along geodesics give rise to the Landsberg curvature L y : Given a Finsler manifold (M, F ), then a global vector field G is induced by F on T M 0 , which in a standard coordinate (x i , y i ) for T M 0 is given by where G i (y) are local functions on T M given by G is called the spray associated to (M, F ). In local coordinates, a curve c(t) is a geodesic if and only if its coordinates (c i (t)) satisfyc i + 2G i (ċ) = 0. For a tangent vector y ∈ T x M 0 , define B y : B and E are called the Berwald curvature and mean Berwald curvature, respectively. Then F is called a Berwald metric and weakly Berwald metric if B = 0 and E = 0, respectively [14]. By definition of Berwald and mean Berwald curvatures, we have

is a family of linear maps on tangent spaces, defined by
∂G j ∂y k . The flag curvature in Finsler geometry is a natural extension of the sectional curvature in Riemannian geometry was first introduced by L. Berwald [3]. For a flag P = span{y, u} ⊂ T x M with flagpole y, the flag curvature K = K(P, y) is defined by K(P, y) := g y (u, R y (u)) g y (y, y)g y (u, u) − g y (y, u) 2 .
When F is Riemannian, K = K(P ) is independent of y ∈ P , and is the sectional curvature of P . We say that a Finsler metric F is of scalar curvature if for any y ∈ T x M , the flag curvature K = K(x, y) is a scalar function on the slit tangent bundle T M 0 . If K = const, then F is said to be of constant flag curvature. A Finsler metric F is called isotropic flag curvature, if K = K(x).
In [1], Akbar-Zadeh considered a non-Riemannian quantity H which is obtained from the mean Berwald curvature by the covariant horizontal differentiation along geodesics. This is a positively homogeneous scalar function of degree zero on the slit tangent bundle. The quantity H y = H ij dx i ⊗dx j is defined as the covariant derivative of E along geodesics [11]. More precisely In local coordinates, we have

. Let (M, F ) be a Finsler manifold. Suppose that the Cartan tensor satisfies in
Proof . Suppose that the Cartan tensor of the Finsler metric F satisfies in Contracting (4) with g ij yields Using (5) and B i h i k = B j h j k = B k , we get I i = (n + 1)B i . Putting this relation in (4), we conclude that F is a C-reducible Finsler metric.

Lemma 3. Let (M, F ) be a Finsler metric. Then F is a GDW-metric if and only if
for some tensor T jkl on manifold M . The geometric meaning of the above identity is that the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic. Proof . Since F satisfies (2), then by considering µ i y i = 0 we get 2E jk = (n + 1)λh ij .

Proof . Let F be is a GDW-metric
On the other hand, we have which implies that 2E jk,l = (n + 1)λ ,l h jk + (n + 1)λ 2C jkl − F −2 (y k h jl + y j h kl ) .
Putting (2), (7) and (8) in (3) yields For the Douglas curvature, we have D i jkl = D i jlk . Then by (9), we conclude that From (9) and (10) we deduce Since F is a Douglas metric, then By Lemmas 2 and 1, it follows that F is a Randers metric.
Proof of Theorem 1. To prove the Theorem 1, we start with the equation (11): Taking a horizontal derivation of (12) implies that where λ ′ = λ |m y m and µ ′ i = µ i|m y m . By Lemma 3, F is a GDW-metric with This completes the proof.
The Funk metric on a strongly convex domain B n ⊂ R n is a non-negative function on T Ω = Ω × R n , which in the special case Ω = B n (the unit ball in the Euclidean space R n ) is defined by the following explicit formula: where | · | and ·, · denote the Euclidean norm and inner product in R n , respectively [14]. The Funk metric on B n is a Randers metric. The Berwald curvature of Funk metric is given by Thus the Funk metric is a GDW-metric which does not satisfy (2). Then by Theorem 1, we conclude the following.

Proof of Theorem 2
To prove Theorem 2, we need the following.
Lemma 4 ( [7,11]). For the Berwald connection, the following Bianchi identities hold: Proof of Theorem 2. We have: Here, we assume that a Finsler metric F is of isotropic flag curvature K = K(x). In local coordinates, R i k = K(x)F 2 h i k . Plugging this equation into (14) gives Differentiating (15) with respect to y m gives a formula for R i jkl,m expressed in terms of K and its derivatives. Contracting (13) with y k , we obtain Multiplying (16) with y i implies that Since F satisfies (2), then we have By contracting (18) with y i , we have By (17) and (19) we get Contracting with g kl yields Since K = 0, then by Deicke's theorem F is a Riemannian metric if and only if µ ′ j = 0.
Theorem 5. Let F be a Finsler metric on an n-dimensional manifold M (n ≥ 3) and satisfies (2). Suppose that F is of scalar flag curvature K. Then K = const if and only if λ ′ = 0.
By taking a horizontal derivative of this equation, we have 2H jk = (n + 1)λ ′ h jk .
Therefore H jk = 0 if and only if λ ′ = 0. By Theorem 4, we get the proof.

Proof of Theorem 3
In this section, we are going to prove Theorem 3.
Proof of Theorem 2. Let F be a Finsler metric satisfy in following where µ i = µ i (x, y) and λ = λ(x, y) are homogeneous functions of degrees −2 and −1 with respect to y, respectively. Contracting (20) with y i yields On the other hand, we have See [14, page 84]. Using (21), (22) and (23), we get By (24), it is obvious that if µ i = 0 then L jkl = 0. Conversely let F be a Landsberg metric. Then we have Contracting (25) with g kl yields µ j = 0. Then F is a Landsberg metric if and only if µ j = 0. Now, contracting (24) with g kl yields J j = − 1 2 (n + 1)F 2 µ j . By using the notion of Landsberg curvature, we define the stretch curvature Σ y : T x M ⊗ T x M ⊗ T x M ⊗ T x M → R by Σ y (u, v, w, z) := Σ ijkl (y)u i v j w k z l where Σ ijkl := 2(L ijk|l − L ijl|k ).
In [3], L. Berwald has introduce the stretch curvature tensor Σ and showed that this tensor vanishes if and only if the length of a vector remains unchanged under the parallel displacement along an infinitesimal parallelogram. Theorem 6. Let (M, F ) be a Finsler manifold on which (2) holds. Suppose that F is a stretch metric. Then µ j is constant along any Finslerian geodesics.
Multiplying (27) with y l implies that By contracting (28) with g jk , we conclude the following (n + 1)µ ′ i = 0. Then on a stretch Finsler spaces, µ i is constant along any geodesics.