Compact affine manifolds with precompact holonomy are geodesically complete

TL;DR

This paper proves that compact affine manifolds with precompact holonomy groups are geodesically complete, establishing a significant link between holonomy properties and geodesic behavior.

Contribution

It demonstrates that the geodesic completeness of compact affine manifolds follows from the precompactness of their holonomy groups, a novel result in differential geometry.

Findings

Compact affine manifolds with precompact holonomy are geodesically complete.

Holonomy group closure being compact implies geodesic completeness.

Provides a new criterion for geodesic completeness in affine geometry.

Abstract

This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.