On a class of Einstein Finsler metrics

Abstract

In this paper, we study a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric and an 1-form. We construct some general (α, β)-metrics with constant Ricci curvature.

Finsler metrics with special Riemannian curvature properties

“…ζ = 0, our proposition has been obtained in [13]. Now we compute the Weyl curvature α W i j of α where α satisfies (3.2).…”

“…In this section, we give a sufficient condition for general (α, β)-metrics of Douglas type to have constant Ricci curvature, generalizing a theorem previously only known in the case of F being projectively equivalent to α [13].…”

“…When F is projectively equivalent to α and satisfies (2.2), i.e., φ 22 − 2(φ 1 − sφ 12 ) = 0 (see Proposition 2.3 (a) in [13]), equivalently, ζ = 0, our proposition has been obtained in [13]. Now we consider Douglas metric F = αφ(b 2 , β α ), where α and β satisfy (3.2) and (2.2) with c 2 > 0.…”

“…Our main approach to show Theorem 1.1 is to determine the Riemannian (or Weyl) curvature of a class of Finsler metrics. By using the formula of Riemannian curvature, we give a sufficient condition for general (α, β)-metrics F = αφ b 2 , β α of Douglas type to have constant Ricci curvature, generalizing a theorem previously only known in the case of F being projectively equivalent to α [13].…”

On a Class of Finsler Metrics With Isotropic Berwald Curvature

“…The main reason is that the category of (α, β) metrics is a little small. If we search Einstein metrics in general (α, β)-metrics, then it is not hard to find out metrics with positive and negative Ricci constant [14]. The classification of projective general (α, β)-metrics with constant flag curvature has just been completed recently by the author and C. Yu [18].…”