On conformally flat (\alpha,\beta)-metrics with relatively isotropic mean Landsberg curvature

Abstract

In this paper, we study conformally flat (α, β)-metrics in the form of F = αφ(β/α), where α is a Riemannian metric and β is a 1-form on the manifold. We prove that conformally flat weak Landsberg (α, β)-metrics must be either Riemannian metrics or locally Minkowski metrics. Further, we prove that, if φ(s) is a polynomial in s, then conformally flat (α, β)-metrics with relatively isotropic mean Landsberg curvature must also be either Riemannian metrics or locally Minkowski metrics.Mathematics Subject Classification: 53B40, 53C60.

On Landsberg general ( α , β )-metrics with a conformal 1-form

“…A Finsler metric is called a weak Landsberg metric if L jkl = 0. Some results about weak landsberg metric can be found in [4,9]. The geodesic coefficients G i of a general (α, β)-metric F = αφ(b 2 , β α ) were given in [20] as the following…”

On the Second Approximate Matsumoto Metric

“…In [7], it is proved that the condition Φ = 0 characterizes the Riemannian metrics among (α, β)-metrics. Hence, in the continue, we suppose that Φ = 0.…”

Randers Metrics of Weakly Isotropic Flag Curvature

“…For an (α, β)-metric F = αφ(s), s = β/α, we have the formulas for the mean Landsberg curvature ( [12]) and the mean Cartan torsion ( [8]) of F as follows…”