# Turkish Journal of Mathematics

## DOI

10.3906/mat-1805-130

## Abstract

The non-Riemannian quantity ${\bf H}$ was introduced by Akbar-Zadeh to characterization of Finsler metrics of constant flag curvature. In this paper, we study two important subclasses of Finsler metrics in the class of so-called $(\alpha,\beta)$-metrics, which are defined by $F=\alpha\phi(s)$, $s=\beta/\alpha$, where $\alpha$ is a Riemannian metric and $\beta$ is a closed 1-form on a manifold. We prove that every polynomial metric of degree $k\geq 3$ and exponential metric has almost vanishing ${\bf H}$-curvature if and only if ${\bf H}=0$. In this case, $F$ reduces to a Berwald metric. Then we prove that every Einstein polynomial metric of degree $k\geq 3$ and exponential metric satisfies ${\bf H}=0$. In this case, $F$ is a Berwald metric.

## Keywords

Polynomial metrics, exponential metric, almost vanishing ${\bf H}$-curvature.

## First Page

207

## Last Page

222

## Recommended Citation

TAYEBI, AKBAR and RAZGORDANI, MASOOME
(2020)
"On ${\bf H}$-curvature of $(\alpha,\beta)$-metrics,"
*Turkish Journal of Mathematics*: Vol. 44:
No.
1, Article 14.
https://doi.org/10.3906/mat-1805-130

Available at:
https://journals.tubitak.gov.tr/math/vol44/iss1/14