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1. CJM 2009 (vol 62 pp. 52)
An Algebraic Approach to Weakly Symmetric Finsler Spaces In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the non-Riemannian reversible weakly
symmetric Finsler spaces we find are non-Berwaldian and with vanishing
S-curvature. This means that reversible non-Berwaldian Finsler spaces
with vanishing S-curvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature Categories:53C60, 58B20, 22E46, 22E60 |
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 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 |