Numerical properties of Koszul connections

TL;DR

This paper investigates the existence, foliation properties, and invariance measures of affine and symplectic structures on Lie groups, addressing open questions in geometric structure theory.

Contribution

It provides new insights into the properties and existence questions of affine and symplectic structures on Lie groups, including invariant and bi-invariant cases.

Findings

Results on existence of affine and symplectic structures on Lie groups.

Characterization of invariant structures and their deviations.

Analysis of foliation properties related to these structures.

Abstract

We use the notation EX(S>M), EXF(S>M) and DL(S>M), where M is a smooth manifold and S is a geometric structure. EX(S>M) is the question whether S exists in M. EXF(S>M) is the question whether M admits S-foliations. DL(S>M) is the search of an invariant measuring how M is far from admitting S. For many major geometric structures, those questions are widly open. In this paper, we address EX(S>M), EXF(S>M) and DL(S>M) for affine structure and symplectic structure, left invariant affine structure, left invariant symplectic structure and bi-invariant riemanniann structure in Lie groups