λ-Projectively Related Finsler Metrics and Finslerian Projective Invariants

   In this paper, by using the concept of spherically symmetric Finsler metric, we define the notion of -projectively related metrics as an extension of projectively related metrics. We construct some non-trivial examples of -projectively related metrics. Let F and   be two -projectively related metrics on a manifold M.  We find the relation between the geodesics of F and   and prove that any geodesic of  F is a multiple of a geodesic of   and the other way around. There are several projective invariants of Finsler metrics, namely, Douglas metrics, Weyl metrics and generalized Douglas-Weyl curvature. We prove that the Douglas metrics, Weyl metrics and generalized Douglas-Weyl metrics are -projective invariants.

First we obtain the spray coefficients of a spherically symmetric Finsler metric. By considering it, we define -projectively related metrics which is a generalization of projectively related Finsler metrics. Then we find the geodesics of two -projectively related metrics. We obtain the relation between Douglas, Weyl and generalized Douglas-Weyl curvatures  of two -projectively related metrics.

We find the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature of two -projectively related Finsler metrics. These calculations tell us that these class of Finsler metrics are -projective invariants.

The following conclusions were drawn from this research. We prove that the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature are -projective invariants. Let F and   be two -projectively related metrics on a manifold M.  We show that F is a Berwald metric if and only if  is a Berwald metric..