A Local Classification of Some Special -Metrics of Constant Flag Curvature

We classify some special Finsler metrics of constant flag curvature on a manifold of dimension . 1. Introduction One of the important problems in Finsler geometry is to study and characterize Finsler metrics of constant flag curvature, which is the generalization of sectional curvature in Riemannian geometry. The local structure of Finsler metrics of constant flag curvature has been historically mysterious and their classification seems to be far from being solved. The -metrics are an important class of Finsler metrics including Randers metrics as the simplest class. By making use of navigation problem, Bao et al. gave a local classification of Randers metrics with constant flag curvature [1]. Recently, Zhou has classified that the -metrics with constant flag curvature in the following form, are locally projectively flat [2]. Lately, Shen and Zhao have studied projectively flat -metrics where is a constant and is a nonzero constant, and they proved such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian [3]. Hence, one natural problem is to consider the classification of such metrics with constant flag curvature. In this paper, we prove the following rigidity result. Theorem 1. Let , where the 1-form is nonzero, is a constant, and is a nonzero constant, be an -metric on a manifold of dimension . Suppose that is of constant flag curvature; then it must be locally Minkowskian. 2. Preliminaries Let be a Finsler metric on an -dimensional manifold and the geodesic coefficients of , which are defined by For any and , the Riemann curvature is defined by -metrics were first introduced by Matsumoto [4]. They are expressed in the following form: where is a Riemannian metric and is a 1-form. is a smooth positive function satisfying For any flag , where , the flag curvature is defined by When is Riemannian, is independent of . It is just the sectional curvature of in Riemannian geometry. is said to be of scalar curvature if, for any , the flag curvature is independent of containing that is equivalent to the following system of equations in a local coordinate system in , If is a constant, then is said to be of constant flag curvature. Let where “ ” denotes the covariant derivative with respect to the Levi-Civita connection of . Clearly is closed if and only if . Moreover, we denote where and . Let and be the geodesic coefficients of and , respectively. Then we have the following. Lemma 2 (see [5]). For an -metric , , the geodesic coefficients are given by where Here and . Let , ; one has the following. Proposition 3 (see [6]).