Ambrose–Singer Connections and Homogeneous Spaces
Homogeneous and locally homogeneous spaces are among the most important objects of study in Differential Geometry. They have been extensively investigated using several methods and techniques. When considering a homogeneous space, many geometric properties translate into algebraic properties. However, a difficulty arises, due to the fact that the same pseudo-Riemannian manifold (M, g) can admit several different descriptions as a coset space G / H. It is surprising how little is understood about this problem for many well-known examples of homogeneous pseudo-Riemannian manifolds.