1. CJM 2012 (vol 65 pp. 621)
||On Surfaces in Three Dimensional Contact Manifolds|
In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all
graphs in the Heisenberg group with vanishing COT. This is
accomplished by showing that the equation for graphs with zero COT can
be decomposed into two first order PDEs, one of which is the backward
invisicid Burgers' equation. Finally we show that the p-minimal graph
equation in the Heisenberg group also has such a
decomposition. Moreover, we can use this decomposition to write down
an explicit formula of a solution near a regular point.
Keywords:contact manifolds, subriemannian manifolds, surfaces
2. CJM 2009 (vol 61 pp. 1201)
||Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups |
A Riemannian manifold $(M,\rho)$ is called Einstein if the metric
$\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some
constant $c$. This paper is devoted to the investigation of
$G$-invariant Einstein metrics, with additional symmetries,
on some homogeneous spaces $G/H$ of classical groups.
As a consequence, we obtain new invariant Einstein metrics on some
Stiefel manifolds $\SO(n)/\SO(l)$.
Furthermore, we show that for any positive integer $p$ there exists a
Stiefel manifold $\SO(n)/\SO(l)$
that admits at least $p$
$\SO(n)$-invariant Einstein metrics.
Keywords:Riemannian manifolds, homogeneous spaces, Einstein metrics, Stiefel manifolds