# Some curvature properties of Cartan spaces with mth root metrics

## TL;DR

This paper investigates curvature properties of m-th root metrics in Cartan geometry, establishing conditions under which these metrics exhibit specific curvature behaviors and transformations.

## Contribution

It provides new conditions for m-th root metrics to be weakly Berwald, weakly Landsberg, and locally Minkowskian under conformal $eta$-changes.

## Key findings

- m-th root metrics with isotropic mean Berwald curvature are weakly Berwald
- m-th root metrics with isotropic mean Landsberg curvature are weakly Landsberg
- conformal $eta$-change conditions lead to locally Minkowskian metrics

## Abstract

In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal $\beta$-change of an m-th root metric be locally dually flat. Finally, we prove that the conformal $\beta$-change of locally projectively flat m-th root metrics are locally Minkowskian.