1. 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
2. CMB 2013 (vol 57 pp. 870)
|A Short Note on Short Pants|
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.
Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22
3. CMB 2011 (vol 54 pp. 396)
|Parabolic Geodesics in Sasakian $3$-Manifolds|
We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.
Keywords:parabolic geodesics, pseudo-Hermitian geometry, Sasakian manifolds
4. CMB 2005 (vol 48 pp. 340)
|Short Geodesics of Unitaries in the $L^2$ Metric |
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$
Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.
Categories:46L51, 58B10, 58B25