On generalized Douglas–Weyl (α, β)-metrics

Abstract

In this paper, we study generalized Douglas-Weyl (α, β)-metrics. Suppose that a regular (α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.

On the Class of Einstein Exponential-Type Finsler Metrics

“…where [18,20,21,23]. Among the (α, β)metrics, the Randers metric F = α + β is a special and significant metric which has important applications in physics, biology, etc.…”

“…If c = 0, then F is a Landsberg metric which is not Berwaldian. In this case, F is a unicorn metric [20]. If c = 0, then F reduces to a Berwald metric.…”

On a class of stretch metrics in Finsler Geometry

“…Therefore, β is a closed 1-form. Since F is a regular (α, β)-metric, (20) implies that F is a Berwald metric.…”

“…Suppose that F is not a Finsler metric of Randers-type. In [20], it is proved that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if φ is given by (1).…”

Unicorn metrics with almost vanishing ${ H}$- and ${ \Xi}$-curvatures

“…If c ̸ = 0 , then F is a Landsberg metric that is not Berwaldian. In this case, F is a unicorn metric [16]. If c = 0 , then F reduces to a Berwald metric.…”