Remark 2

This situation presents itself in particular when G is a real Lie group, of finite dimension, countable at infinity, operating transitively and analytically on an analytic manifold X. If one takes for H the stabiliser of a point of X, the mapping $p$ is a submersion of G onto the homogeneous Lie space $G/H$, isomorphic to X (LIE, III, p. 109, corollary). It therefore possesses local sections (VAR, p. 50).