User:Rybu/geometric manifold

A geometric manifold is a manifold $$M$$ which has a geometric structure. To make sense of the definition of a geometric structure, a construction from the subject of Lie Groups is helpful.

A geometric structure is a complete Riemann metric on $$M$$ (if $$M$$ has boundary, then the convention is the Riemann metric need only be defined on the interior of $$M$$) such that its universal cover $$\tilde M$$ (with the induced Riemann metric) is isometric to a homogeneous space where we assume the homogeneous space has its natural Riemann metric. This of course requires the homogeneous space to satisfy that its point stabilizers are compact.