On general (α,β)-metrics of Landsberg type

Abstract

In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric [Formula: see text] and a one-form [Formula: see text]. They are called general [Formula: see text]-metrics. We have proven that, every Landsberg general [Formula: see text]-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general [Formula: see text]-metrics.

On general $(α, β)$-metrics of weak Landsberg type

“…Proof of Theorem 1.Since in [11] is obtained exactly (20) and (21) for Landsberg metrics, according to 4, it is obvious.…”

“…Proof of Corollary 1. Here, by [11], we have E −sE 2 = 0 and H 2 −sH 22 = 0 also. In [10], it is proved that a general (α, β)-metric where β is closed and conformal one-form, is a Berwald metric if and only if…”

“…where E and H is defined in (11) and ( 12), l j := a ij l i and (k → l → j → k) denotes cyclic permutation.…”

“…3 The mean Landsberg curvature of General (α, β)metrics By use of Proposition 1, M. Zohrehvand and H. Maleki calculated the Landsberg curvature of a general (α, β)-metric in [11], when β is a closed and conformal one-form:…”

“…In general (α, β)-metrics, Zohrehvand and Maleki in [11] showed that hunting for an unicorn cannot be successful in the class of metrics where β is a closed and conformal one-form, i.e. b i|j = ca ij , where b i|j is the covariant derivation of β with respect to α and c = c(x) is a scalar function on M .…”