# On $S$-Curvature of Homogeneous Finsler spaces with $(\alpha, \beta)$-metrics

## TL;DR

This paper investigates the curvature properties of homogeneous Finsler spaces with specific $( ext{alpha}, eta)$-metrics, deriving explicit formulas for $S$-curvature and mean Berwald curvature, and proving the existence of invariant vector fields.

## Contribution

It provides new explicit formulas for $S$-curvature and demonstrates the existence of invariant vector fields on certain homogeneous Finsler spaces with $( ext{alpha}, eta)$-metrics.

## Key findings

- Explicit formula for $S$-curvature of $( ext{alpha}, eta)$-metrics.
- Existence of invariant vector fields on these spaces.
- Calculation of mean Berwald curvature using the $S$-curvature formula.

## Abstract

The study of curvature properties of homogeneous Finsler spaces with $(\alpha, \beta)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(\alpha, \beta)$-metric and Randers changed square $(\alpha, \beta)$-metric is proved. Further, an explicit formula for $S$-curvature of these $(\alpha, \beta)$-metrics is established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(\alpha, \beta)$-metrics is calculated.