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Search: MSC category 53B40 ( Finsler spaces and generalizations (areal metrics) )

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1. CJM 2012 (vol 65 pp. 1401)

Zhao, Wei; Shen, Yibing
A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results
In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger-Kazdan inequality and Santaló's formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

Keywords:Finsler manifold, Berger-Kazdan inequality, Berger-Kazdan comparison theorem, Santaló's formula, Croke's isoperimetric inequality
Categories:53B40, 53C65, 52A38

2. CJM 2009 (vol 61 pp. 1357)

Shen, Zhongmin
On a Class of Landsberg Metrics in Finsler Geometry
In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a $1$-form on a manifold. We show that a \emph{regular} Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, $\phi=\phi(s)$, for which there are a Riemannian metric $\alpha$ and a $1$-form $\beta$ on a manifold $M$ such that the scalar function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.

Categories:53B40, 53C60

3. CJM 2008 (vol 60 pp. 443)

Shen, Z.; Yildirim, G. Civi
On a Class of Projectively Flat Metrics with Constant Flag Curvature
In this paper, we find equations that characterize locally projectively flat Finsler metrics in the form $F = (\alpha + \beta)^2/\alpha$, where $\alpha=\sqrt{a_{ij}y^iy^j}$ is a Riemannian metric and $\beta= b_i y^i$ is a $1$-form. Then we completely determine the local structure of those with constant flag curvature.


4. CJM 2003 (vol 55 pp. 112)

Shen, Zhongmin
Finsler Metrics with ${\bf K}=0$ and ${\bf S}=0$
In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

Categories:53C60, 53B40

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