1. CMB Online first
|Ricci Curvature Tensor and Non-Riemannian Quantities|
There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. By this Ricci curvature tensor, we shall have a better understanding on these non-Riemannian quantities.
Keywords:Finsler metrics, sprays, Ricci curvature, non-Riemanian quantity
2. CMB 2011 (vol 56 pp. 184)
|On Some Non-Riemannian Quantities in Finsler Geometry|
In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the S-curvature. We show some relationships among the flag curvature, the S-curvature, and the new non-Riemannian quantity.
Keywords:Finsler metric, S-curvature, non-Riemannian quantity
3. CMB 2011 (vol 56 pp. 615)
|Randers Metrics of Constant Scalar Curvature|
Randers metrics are a special class of Finsler metrics. Every Randers metric can be expressed in terms of a Riemannian metric and a vector field via Zermelo navigation. In this paper, we show that a Randers metric has constant scalar curvature if the Riemannian metric has constant scalar curvature and the vector field is homothetic.
Keywords:Randers metrics, scalar curvature, S-curvature
4. CMB 2011 (vol 55 pp. 474)
|A Note on Randers Metrics of Scalar Flag Curvature|
Some families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic $S$-curvature are given. Certain Randers metrics with Einstein $\alpha$ are considered and proved to be complex. Three dimensional Randers manifolds, with $\alpha$ having constant scalar curvature, are studied.
Keywords:Randers metrics, scalar flag curvature
5. CMB 2009 (vol 52 pp. 132)
|On Projectively Flat $(\alpha,\beta)$-metrics |
The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $(\alpha,\beta)$-metrics defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n \geq 3$.
6. CMB 2005 (vol 48 pp. 112)
|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