S-curvature of isotropic Berwald metrics

Abstract

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.

On a Class of Finsler Metrics With Isotropic Berwald Curvature

“…By Theorem 4.2, we obtain that F is either a Berwald metric given by (1.6) or a Randers metric expressed by (1.7). By Theorem 1.1 in [16], F is of isotropic S-curvature. From Theorem 5.2, f , g and h in (1.7) satisfy (1.8).…”

“…Finally, it is worth mentioning that the S-curvature is an important nonRiemannian quantity in Finsler geometry [4,7,16]. It interacts with the flag curvature in a mysterious way.…”

“…S(x, y) is called the S-curvature [7,9,16]. F is said to have isotropic S-curvature if there is a scalar function κ(x) on M such that (2.5)…”

“…A Finsler metric F on a manifold M is said to be of isotropic Berwald curvature if its Berwald curvature B j i kl satisfies Chen-Shen showed that a Finsler metric F is of isotropic Berwald curvature if and only if it is a Douglas metric with isotropic mean Berwald curvature [6]. Tayebi-Rafie's result tells us that every isotropic Berwald metric is of isotropic S-curvature [16]. In [15], Tayebi-Najafi proved that isotropic Berwald metrics of scalar flag curvature are of Randers type.…”

On a Class of Locally Dually Flat Isotropic Berwald Metrics

“…Lemma 2 [5]. Let F be an isotropic Berwald metricSuppose that G i are the geodesic spray coefficients of F satisfy…”