On Cartan Spaces with m-th Root Metrics

TL;DR

This paper investigates Cartan spaces with m-th root metrics, showing that certain isotropic curvature conditions simplify these spaces to well-known types and establishing conditions for vanishing H-curvature.

Contribution

It introduces new curvature properties for Cartan spaces with m-th root metrics and proves reduction results under isotropic curvature conditions.

Findings

Isotropic Landsberg curvature implies Landsberg space

Isotropic mean Landsberg curvature implies weakly Landsberg space

Almost vanishing H-curvature leads to H=0

Abstract

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg and weakly Berwald space, respectively. Then we show that m-th root Cartan space of almost vanishing H-curvature satisfies H = 0.