Post-Lie Algebra Structure of Manifolds with Constant Curvature and Torsion

TL;DR

This paper explores the algebraic structure of vector fields on manifolds with constant curvature and torsion, extending previous flat connection results to more general cases and applying this to differential equations.

Contribution

It introduces a post-Lie algebra framework for manifolds with non-flat connections, incorporating holonomy endomorphisms, and details the universal Lie algebra structure.

Findings

Post-Lie algebra structure exists for affine connections with parallel torsion and curvature.

In non-flat cases, holonomy endomorphisms are essential for the algebraic structure.

Applications include new methods for solving differential equations on such manifolds.

Abstract

For a general affine connection with parallel torsion and curvature, we show that a post-Lie algebra structure exists on its space of vector fields, generalizing previous results for flat connections. However, for non-flat connections, the vector fields alone are not enough, as the presence of curvature also necessitates that we include endomorphisms corresponding to infinitesimal actions of the holonomy group. We give details on the universal Lie algebra of this post-Lie algebra and give applications for solving differential equations on manifolds.