User:Rybu/geometric manifold

From Wikipedia, the free encyclopedia

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 (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.