On Randers Metrics with Isotropic S-Curvature

In this paper, we study a class of Finsler metrics defined by a vector field on a Riemannian space form. We give an explicit formula for those with isotropic S-curvature. This class contains all Randers metrics of constant flag curvature.

Homogeneous manifolds admitting non-Riemannian Einstein-Randers metrics

In this paper, we find some new homogeneous manifolds G/H admitting non-Riemannian EinsteinRanders metrics when G is the compact simple Lie group E6, or E7 or E8. In the beginning, we prove that these homogeneous manifolds admit Riemannian Einstein metrics. Based on these metrics, we obtain non-Riemannian Einstein Randers metrics on them.

PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

This article characterizes projectively fiat Finsler metrics with almost isotropic S-curvature.

A new quantity in Finsler geometry

In this paper,we study a new Finslerian quantity■defined by the T-curvature and the angular metric tensor.We show that the■-curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag curvature but also has a vanishing trace.We find that the■-curvature is closely related to the Riemann curvature,the Matsumoto torsion and theΘ-curvature.We solve Z.Shen’s open problem in terms of the■-curvature.Finally,we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the■-curvature,generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature.

On conformal vector fields on Randers manifolds

In this paper,we study conformal vector fields on a Randers manifold with certain curvature properties.In particular,we completely determine conformal vector fields on a Randers manifold of weakly isotropic flag curvature.

On the Ricci Curvature of a Randers Metric of Isotropic S-curvature

We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.

On the projective Ricci curvature

The notion of the Ricci curvature is defined for sprays on a manifold.With a volume form on a manifold,every spray can be deformed to a projective spray.The Ricci curvature of a projective spray is called the projective Ricci curvature.In this paper,we introduce the notion of projectively Ricci-flat sprays.We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature.Then we study and characterize projectively Ricci-flat Randers metrics.

On a class of two-dimensional Finsler manifolds of isotropic S-curvature

For an(α, β)-metric(non-Randers type) of isotropic S-curvature on an n-dimensional manifold with non-constant norm ‖β‖α, we first show that n = 2, and then we characterize such a class of two-dimensional(α, β)-manifolds with some PDEs, and also construct some examples for such a class.