%0 Journal Article %T Reductive locally homogeneous pseudo-Riemannian manifolds and Ambrose-Singer connections %A Ignacio Luj¨˘n %J Mathematics %D 2014 %I arXiv %X 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. %U http://arxiv.org/abs/1405.0826v1