Torsion-free Connections with prescribed Curvature

Abstract

We provide necessary and sufficient conditions for a curvature map $S\colon U\longrightarrow K(\mathfrak{g})$ to arise from the curvature tensor of a torsion-free connection on a sufficiently small $U'\subset U$ by using a suitable power series approach. This torsion-free connection is uniquely given. The curvature map $S$ is used to effectively compute the holonomy of this connection.