On the Second Approximate Matsumoto Metric

Abstract

Abstract. In this paper, we study the second approximate Matsumoto metric F = α + β + β 2 /α + β 3 /α 2 on a manifold M . We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

Matsumoto Metrics of Reversible Curvature

“…By (15), it follows that α 2 divides A 0 . Since α 2 is an irreducible polynomial in y, it must be the case that α 2 divides r 00 .…”

“…The class of (α, β)-metrics was introduced by Matsumoto as extension of Randers and Kropina metrics [9]. An (α, β)-metric is a Finsler metric on M defined by F := αφ(s), where s = β/α, φ = φ(s) is a C ∞ function on the (−b 0 , b 0 ) with certain regularity, α is a Riemannian metric and β is a 1-form on M [3][14] [15].…”

“…This metric was introduced by Matsumoto as a realization of Finsler's idea "a slope measure of a mountain with respect to a time measure" [12] [13]. He gave an exact formulation of a Finsler surface to measure the time on the slope of a hill and introduced the Matsumoto metrics in [8] [15].…”