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1. CJM 2012 (vol 65 pp. 1401)
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
SantalÃ³'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, SantalÃ³'s formula, Croke's isoperimetric inequality Categories:53B40, 53C65, 52A38 |
2. CJM 2009 (vol 61 pp. 1357)
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)
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.
Category:53B40 |
4. CJM 2003 (vol 55 pp. 112)
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 |