New Characterization of Flat Affine Manifolds and the Associative Envelope of a Left Symmetric Algebra

TL;DR

This paper introduces a new way to characterize flat affine connections on real manifolds using affine representations, and explores their implications for Lie groups and affine structures.

Contribution

It provides a novel characterization of flat affine connections via affine representations and examines their properties on Lie groups and manifolds.

Findings

Existence of Lie groups with flat affine bi-invariant connections

Characterization of flat affine connections through affine representations

Examples illustrating the theoretical results

Abstract

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connection on a Lie group. In the last case, we show the existence of a Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains the Lie algebra of complete infinitesimal affine transformations of the given Lie group. We also prove some results about flat affine manifolds whose group of diffeomorphisms admit a flat affine bi-invariant structure. The paper is illustrated with several examples.