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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.
Field : Fen Bilimleri ve Matematik
Journal Type : Uluslararası
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