A GEOMETRIC CHARACTERIZATION OF FINSLER MANIFOLDS OF CONSTANT CURVATURE K = 1

We prove that a Finsler manifold Fm is of constant curvature K = 1 if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IM of Fm.


Introduction.
The geometry of Finsler manifolds of constant curvature is one of the fundamental subjects in Finsler geometry.Akbar-Zadeh [1] proved that, under some conditions on the growth of the Cartan tensor, a Finsler manifold of constant curvature K is locally Minkowskian if K = 0 and Riemannian if K = −1.Recently, Bryant [5] has constructed interesting Finsler metrics of positive constant curvature on the sphere S 2 .Shen [9] has also investigated the geometric structure of Finsler manifolds of positive constant curvature via the Riemannian Y -metrics.Some special Finsler metrics of constant curvature have been intensively studied by Matsumoto [7,8], Shibata-Kitayama [10], and Wei [11].
The purpose of the present paper is to obtain a geometric characterization of Finsler manifolds of positive constant curvature.More precisely, we prove that F m = (M, F ) is a Finsler manifold of constant curvature K = 1 if and only if the unit horizontal Liouville vector field ξ = (y i /F )δ/δx i is a Killing vector field on the indicatrix bundle IM of F m .To achieve this result, we consider the Sasaki-Finsler metric G on T M and prove that the linear connection of the Cartan connection on F m is just the projection of the Levi-Civita connection ∇ with respect to G on the vertical vector bundle (see Theorem 2.1).This enables us to express the local coefficients of ∇ in terms of the local coefficients of the Cartan connection of F m (see Theorem 2.2).Finally, a necessary and sufficient condition for ξ to be a Killing vector field on IM leads to the proof of the main result stated in Theorem 3.3.

2.
The Levi-Civita connection with respect to a Sasaki-Finsler metric.In the present section, we show that the linear connection of the Cartan connection is the projection of the Levi-Civita connection with respect to the Sasaki-Finsler metric on the vertical vector bundle.Then we express the local coefficients of the Levi-Civita connection in terms of the local coefficients of the Cartan connection.
Throughout the paper we use the Einstein convention, that is, repeated indices with one upper index and one lower index denotes summation over their range.Also, for any smooth manifold N, we denote by Ᏺ(N) the algebra of smooth functions on N and by Γ (E) the Ᏺ(N)-module of smooth sections of a vector bundle E over N.For some Finsler tensor fields we put the index o to denote the contraction by the supporting element y i , as for example, T io = T ij y j .
Let F m = (M, F ) be a Finsler manifold, where M is a real m-dimensional smooth manifold and F is the fundamental function of F m (see Antonelli-Ingarden-Matsumoto [2, page 36]).Consider T M • = T M\{0} and denote by V T M • the vertical vector bundle over T M • , that is, V T M • = ker π * , where π * is the tangent mapping of the canonical projection π : T M • → M. We may think of the Finsler metric g = (g ij (x, y)), where we set as a Riemannian metric on V T M • .The canonical nonlinear connection HT M • = (N j i (x, y)) of F m is given by ) (2.2b) Then on any coordinate neighborhood ᐁ ⊂ T M • the vector fields where we set On T M • we consider the almost product structure Q locally given by Then by means of the pair (g, Q), we define a Riemannian metric G on T M • by (cf. Bejancu [4, page 42]) where v and h denote the projection morphisms of T T M • on V T M • and HT M • , respectively.Clearly, we have that is, HT M • and V T M • are complementary orthogonal vector subbundles of T T M • with respect to G.As the Riemannian metric G is of Sasaki type and was obtained from a Finsler metric, we call it the Sasaki-Finsler metric on T M • .The Levi-Civita connection ∇ on T M • with respect to G is given by the well-known formula On the other hand, the Cartan connection of F m is the pair FC = (HT M • , ∇ • ), where HT M • is the canonical nonlinear connection given by (2.2) and ∇ • is a linear connection on V T M • whose local coefficients C i k j and F i k j are given by (2.11a) and (2.12a) respectively.The h-and v-covariant derivatives of a Finsler tensor field T = (T In order to get an interrelation between the Levi-Civita connection ∇ and the linear connection ∇ • of the Cartan connection we set G j = g jh G h , and by direct calculations using (2.1) and (2.2b), we deduce that (2.13) Now, we state the following result.

Theorem 2.2. The Levi-Civita connection ∇ on T M • with respect to the Sasaki-Finsler metric G is locally expressed in terms of the local coefficients of the Cartan connection of F m as follows:
(2.30) (2.31)

The main result.
It is well known that, on the tangent bundle T M, there exists a globally defined vector field L = y i (∂/∂y i ) called the Liouville vector field.By means of the almost product structure Q, we obtain another vector field QL = y i (δ/δx i ) which we call the horizontal Liouville vector field of F m .Clearly, ξ = i (δ/δx i ), where i = y i /F is a unit vector field with respect to G. To state the next theorem, we recall that the angular metric of F m has the local components Also, we recall that the Lie derivative of G with respect to ξ is given by (cf.Yano-Kon [12, page 41]) Now we prove the following theorem.
Theorem 3.1.The Lie derivative of G with respect to ξ satisfies the equations Proof.First, by using (2.31), (2.9) and taking into account that N k i = y j F i k j , we obtain since N h j are positively homogeneous of degree 1 with respect to (y k ).Next, by using (2.29) and (2.9), we deduce that since k |j = 0. Taking into account (3.2), we see that (3.5) and (3.6) yield (3.3).Finally, substituting X and Y from (3.2) by δ/δx j and ∂/∂y i , respectively, and using (2.29), (2.31), and (2.9), we infer that since by equations (17.30) and (17.21) in Matsumoto [6] we have i j = (1/F )h ij and R ioj = R joi .As (3.7) implies (3.4), the proof is complete.
Next, for a fixed point x ∈ M we consider the indicatrix I x at x, which is a hypersurface of the fibre T M • x given by the equation F(x,y) = 1.Then denote by IM the hypersurface of T M • consisting of indicatrices at all points of M and call it the indicatrix bundle over F m .It is easy to show that Qξ = i (∂/∂y i ) is the unit normal vector field with respect to the Sasaki-Finsler metric.Indeed, if the local equations of IM in T M • are then, we have As the h-covariant derivative of F vanishes, by using (2.3), we obtain The natural frame field on IM is represented by Then by (3.10), we deduce ∂y k ∂u α y h g hk = 0. (3.12) Thus Qξ is orthogonal to any vector tangent to IM.The horizontal Liouville vector field is tangent to IM since G(ξ, Qξ) = 0.
To state the next corollary, we recall that ξ is a Killing vector field on IM with respect to G if and only if L ξ G = 0 (cf.Yano-Kon [12, page 41]).Thus, by Theorem 3.1, we may state the following corollary.

Corollary 3.2. The unit horizontal Liouville vector field ξ is a Killing vector field on the indicatrix bundle IM if and only if
Now, we consider a Finsler vector field X = X i (∂/∂y i ) which is noncolinear to the Liouville vector field L. Then the curvature (flag curvature) of F m for the flag spanned by {L, X} is the function (see Equation (26.1) in Matsumoto [6] or Bao-Cheen-Shen [3]) In case K is a constant we say that F m is a Finsler manifold of constant curvature.The above results enable us to state a geometric characterization of Finsler manifolds of constant curvature by means of the horizontal Liouville vector field, which is the main result of this paper.
Theorem 3.3.The Finsler manifold F m is of constant curvature K = 1 if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IM.
In the above discussions the constant curvature was taken to be K = 1 for "normalisation" purposes only.However the geometric characterization remains valid for any positive constant curvature.To be more precise, we give the following.For any real number λ > 0, we define the λ-indicatrix bundle I λ M to be: (3.17) Then simple modifications in the calculations given earlier will show that the unit horizontal Liouville vector field ξ is a Killing vector field on I λ M if and only if h ij (x, y) = R ioj (x, y), ∀(x, y) ∈ I λ M. So, as before, this can be used to prove the following theorem.
Theorem 3.4.The Finsler manifold F m is of constant positive curvature λ if and only if the unit horizontal Liouville vector field is a Killing vector field on I λ M.
, which together with (2.16) proves the assertion of the theorem.Next, in order to get the local coefficients of ∇, we consider the local frame field {δ/δx i , ∂/∂y i } on T M • and set