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Search: MSC category 53C60 ( Finsler spaces and generalizations (areal metrics) [See also 58B20] )

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1. CMB 2017 (vol 60 pp. 253)

Chen, Bin; Zhao, Lili
On a Yamabe Type Problem in Finsler Geometry
In this paper, a new notion of scalar curvature for a Finsler metric $F$ is introduced, and two conformal invariants $Y(M,F)$ and $C(M,F)$ are defined. We prove that there exists a Finsler metric with constant scalar curvature in the conformal class of $F$ if the Cartan torsion of $F$ is sufficiently small and $Y(M,F)C(M,F)\lt Y(\mathbb{S}^n)$ where $Y(\mathbb{S}^n)$ is the Yamabe constant of the standard sphere.

Keywords:Finsler metric, scalar curvature, Yamabe problem
Categories:53C60, 58B20

2. CMB 2016 (vol 59 pp. 575)

Li, Jifu; Hu, Zhiguang; Deng, Shaoqiang
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
Categories:53C30, 53C60

3. CMB 2015 (vol 58 pp. 530)

Li, Benling; Shen, Zhongmin
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
Categories:53B40, 53C60

4. CMB 2011 (vol 56 pp. 184)

Shen, Zhongmin
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
Categories:53C60, 53B40

5. CMB 2011 (vol 56 pp. 615)

Sevim, Esra Sengelen; Shen, Zhongmin
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
Categories:53C60, 53B40

6. CMB 2011 (vol 55 pp. 474)

Chen, Bin; Zhao, Lili
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
Categories:53B40, 53C60

7. CMB 2009 (vol 52 pp. 132)

Shen, Zhongmin
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 $n \geq 3$.

Categories:53B40, 53C60

8. CMB 2005 (vol 48 pp. 112)

Mo, Xiaohuan; Shen, Zhongmin
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


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