I have one question: Let $G$ is a group Lie and $H$ is closed subgroup. Let $M=\cup g^{-1}Hg < G$. Is it true that $M$ is manifold? What is the dimension of $M$?
upd. Let $G$ is a compact. I have a hypothesis that, in this case $M$ is manifold, and $\dim \, M = \dim \, G + \dim \, H - \dim \, N_G(H) $.
I assume that the union, in the definition of $M$, is over all elements $g$ of $G$. Let $G$ be the group of orientation-preserving rigid motions of the Euclidean plane (i.e., translations and rotations), and let $H$ be the subgroup of rotations about a certain point $P$. Then, unless I've made a mistake, $M$ consists of all rotations about all points, including the identity element of the group, but not including any non-trivial translations. This $M$ is not a manifold, because no neighborhood of the identity looks like Euclidean space.