Xiaohuan Mo

On the Flag Curvature of Finsler Metrics of Scalar Curvature

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.

On Negatively Curved Finsler Manifolds of Scalar Curvature

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n \geq 3. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat

On geodesics of Finsler metrics via navigation problem

This paper is devoted to a study of geodesics of Finsler metrics via Zermelo navigation. We give a geometric description of the geodesics of the Finsler metric produced from any Finsler metric and any homothetic field in terms of navigation representation, generalizing a result previously only known in the case of Randers metrics with constant S-curvature. As its application, we present explicitly the geodesics of the Funk metric on a strongly convex domain.

ON A CLASS OF PROJECTIVELY FLAT FINSLER METRICS OF NEGATIVE CONSTANT FLAG CURVATURE

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shens construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

The explicit construction of all dually flat Randers metrics

In this paper, we construct explicitly all dually flat Randers metrics by using the bijection between Randers metrics and their navigation representation.

ON THE CONSTRUCTION OF DUALLY FLAT FINSLER METRICS

In this paper, we give a new approach to find a dually flat Finsle r m tric. As its application, we produce many new spherically symmetric dua lly flat Finsler metrics by using known projective spherically symmetric Finsler me trics.

On Finsler surfaces of constant curvature with two-dimensional isometry group

In this paper, we study Finsler surfaces of constant (flag) curvature. We show that the space of those, with two-dimensional isometric group depends on two arbitrary constants. We also give a new technique to recover Finsler metrics from the specified two constants. Using this technique we obtain some new Finsler surfaces of constant flag curvature with two-dimensional isometry group.