On the Class of -Metrics

We study a class of Finsler metrics in the form , where is a Riemannian metric, is a -form, and is called -metrics. We find the necessary and sufficient conditions under which the class of -metrics is locally projectively flat and Douglas metrics, respectively. 1. Introduction It is Hilbert’s Fourth Problem in the regular case to study and characterize Finsler metrics on an open domain whose geodesics are straight lines. Finsler metrics with this property are called projectively flat metrics. It is easy to see that a Finsler metric on an open subset is projectively flat if and only if the spray coefficients are in the form , where is a positively homogeneous function of degree one in . In 1903 G. Hamel found a system of partial differential equations that characterize projectively flat Finsler metrics on an open subset . That is, A natural problem is to find projectively flat metrics by solving (1). The Beltrami Theorem tells us that a Riemannian metric is locally projectively flat if and only if it is of constant sectional curvature. Thus this problem has been solved in Reimannian Geometry. However for Finsler metrics, this problem is far from being solved. In order to find examples of projectively flat Finsler metrics, we consider -metrics. An -metric is defined by , , where is a positive scalar function on with certain regularity, is a Riemannian metric, and is a -form on a manifold . The simplest -metric is Randers metric . In [1]; it is proved that a Randers metric on a manifold is locally projectively flat if and only if is projectively flat and is closed. The other important -metric is Berwald metric on a manifold . In [2], Shen and Yildirim find necessary and sufficient conditions under which the Berwald metric is projectively flat. They proved that the Berwald metric is projectively flat if and only if the following conditions hold:(1) ,(2)the spray coefficients of are in the form ,where , denote the covariant derivatives of with respect to , is a scalar function, and is a -form on . Also they determined local structure of with constant flag curvature. In [3, 4], some other special -metrics are studied. In this paper, we consider a special subclass of -metrics in the following form: where and . We call it -metric. We prove the following. Theorem 1. Let be a -metric on an open subset . is locally projectively flat if and only if the following conditions hold:(1) is parallel with respect to ;(2) is locally projectively flat; that is, is of constant curvature. It is said that a Finsler metric is trivial if it satisfies the conclusion of Theorem 1.