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 Online first
||Cohomogeneity one Randers metrics|
An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.
Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metrics
3. 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
4. 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
5. 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
6. 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 $n \geq 3$.
7. 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