  Ambrose and Singer characterized connected, simply-connected and complete
homogeneous Riemannian manifolds as Riemannian manifolds admitting a metric
connection such that its curvature and torsion are parallel. The aim of this
paper is to extend Ambrose-Singer Theorem to the general framework of locally
homogeneous pseudo-Riemannian manifolds. In addition we study under which
conditions a locally homogeneous pseudo-Riemannian manifold can be recovered
from the curvature and their covariant derivatives at some point up to finite
order. The same problem is tackled in the presence of a geometric structure.
