Generalized Isotropic Berwald Manifolds

TL;DR

This paper introduces a new class of Finsler manifolds called generalized isotropic Berwald manifolds, explores their geometric properties, and characterizes their curvature features, extending existing classifications in Finsler geometry.

Contribution

It constructs the generalized isotropic Berwald manifolds, proves their relation to Douglas-Weyl manifolds, and characterizes curvature conditions on these manifolds.

Findings

Every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold.

On compact such manifolds, stretch and Landsberg curvatures are equivalent.

A Finsler metric is R-quadratic iff it is a stretch metric with vanishing E-curvature.

Abstract

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald manifold, we show that the notions of stretch and Landsberg curvatures are equivalent. Then we prove that on these manifolds, a Finsler metric is R-quadratic if and only if it is a stretch metric with vanishing E -curvature. Finally, we determine the flag curvature of generalized isotropic Berwald manifold with scalar flag curvature.