Torsion-free Connections with prescribed Curvature

TL;DR

This paper characterizes when a curvature map can originate from a unique torsion-free connection, providing conditions and a method to compute the connection's holonomy using a power series approach.

Contribution

It establishes necessary and sufficient conditions for curvature maps to come from torsion-free connections and introduces a power series method for explicit construction and holonomy computation.

Findings

Derived conditions for curvature maps to correspond to torsion-free connections

Provided a power series method for constructing such connections

Enabled effective computation of the connection's holonomy

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.