On $(\alpha ,\beta ,\gamma )$-metrics

Abstract

In this paper, a new class of Finsler metrics which are included $(\alpha ,\beta )$-metrics are introduced. They are defined by a Riemannian metric and two 1-forms $\beta =b_i(x)y^i$ and $\gamma ={\gamma }_i(x)y^i$. This class of metrics are a generalization of $(\alpha ,\beta )$-metrics which are not always $(\alpha ,\beta )$-metric. We find a necessary and sufficient condition for this metric to be locally projectively flat and then we prove the conditions for this metric to be of Douglas type.

arXiv comment19 pages