1. CMB 2016 (vol 59 pp. 575)
||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 2011 (vol 54 pp. 244)
||Homogeneous Suslinian Continua|
A continuum is said to be Suslinian if it does not
contain uncountably many
mutually exclusive non-degenerate subcontinua. Fitzpatrick and
Lelek have shown that a metric Suslinian continuum $X$ has the
property that the set of points at which $X$ is connected im
kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua
and obtain a number of corollaries. In particular, we prove that a homogeneous,
non-degenerate, Suslinian continuum is a simple closed curve and that each separable,
non-degenerate, homogenous, Suslinian continuum is metrizable.
Keywords:connected im kleinen, homogeneity, Suslinian, locally connected continuum
Categories:54F15, 54C05, 54F05, 54F50
3. CMB 2010 (vol 53 pp. 412)
||Einstein-Like Lorentz Metrics and Three-Dimensional Curvature Homogeneity of Order One|
We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.
Keywords:Lorentz manifolds, curvature homogeneity, Einstein-like metrics
Categories:53C50, 53C20, 53C30