Geometric Meanings of Curvatures in Finsler Geometry

In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.Comment: 17 pages, This article is for the 20th Winter School on Geometry and Physics at Srni in Czech Republic. It was written during my visit at Institute of Mathematics and Informatics (IMI) at University of Debrecen in Hungary. The author would like to thank Dr. S. B\'{a}cs\'{o} and Dr. L. Kozma for their great help and hospitalit

On a class of projectively flat Finsler metrics

In this paper, we study a class of Finsler metrics composed by a Riemann metric $\alpha=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $\beta=b_i(x)y^i$ called general ($\alpha$, $\beta$)-metrics. We classify those projectively flat when $\alpha$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of projectively flat Finsler metrics is found

On a class of Einstein Finsler metrics

In this paper, we study a class of Finsler metrics called general (\alpha,\beta)-metrics, which are defined by a Riemannian metric and an 1-form. We construct some general (\alpha,\beta)-metrics with constant Ricci curvature.Comment: 14 pages, no figur