Reductive locally homogeneous pseudo-Riemannian manifolds and Ambrose-Singer connections

TL;DR

This paper extends the Ambrose-Singer theorem to locally homogeneous pseudo-Riemannian manifolds, exploring conditions under which these manifolds can be reconstructed from curvature data, including cases with additional geometric structures.

Contribution

It generalizes the Ambrose-Singer characterization from Riemannian to pseudo-Riemannian settings and investigates reconstruction conditions with geometric structures.

Findings

Extension of Ambrose-Singer theorem to pseudo-Riemannian manifolds

Conditions for manifold reconstruction from curvature derivatives

Analysis of geometric structures in the reconstruction process

Abstract

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.