Search

In this paper, we study a class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics, which are defined by a Riemannian metric and a 1-form. These metrics form a rich class of Finsler metrics. Suppose that $$F=\alpha \phi (b^{2},s)$$ is a general $$(\alpha ,\beta )$$ -metric which is conformally related to $$\tilde{F}=e^{\sigma (x)}F,$$ where $$\sigma := \sigma (x)$$ is a scalar function on M. We show that if F is Douglas metric, then $$\tilde{F}$$ is also Douglas metric if and only if the conformal transformation is a homothety under suitable condition on $$\phi $$ . Furthermore, we have investigated the conformal transformation of isotropic S-curvature general $$(\alpha ,\beta )$$ -metrics.

Finsler metrics of quadratic curvatures form a rich and important class of metrics. In this paper, we study spherically symmetric metrics of quadratic curvatures. We find equations that characterize R-quadratic spherically symmetric Finsler metrics. In addition, we find the necessary and sufficient conditions for the metrics to be W-quadratic and Ricci-quadratic types.

In this paper, we study $$\Xi $$ -curvature and H-curvature of a special class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics. We prove that every general $$(\alpha ,\beta )$$ -metric of almost vanishing H-curvature is of almost vanishing $$ \Xi $$ -curvature under certain conditions. Moreover, we study such Finsler metric with vanishing $$\Xi $$ -curvature and its interaction to the flag curvature.

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.

Projective Ricci curvature is a projective invariant quantity in Finsler geometry which is introduced by Z. Shen. In this paper, we study special projective Ricci curvature of C-reducible Finsler metrics. The necessary and sufficient conditions of these metrics, which cause these metrics to be weak or isotropic projective Ricci curvature, are found and it is proved that C-reducible Douglas metric of isotropic PRic-curvature must be PRic flat. The same theorem for C-reducible metrics of scalar flag curvature is also investigated.