Connection preserving actions are topologically engaging

TL;DR

This paper demonstrates that actions of simple noncompact Lie groups preserving certain geometric structures are topologically engaging on a large subset of the manifold, aiding in understanding rigidity phenomena.

Contribution

It establishes that such group actions are topologically engaging on an open conull dense set, extending rigidity results to broader geometric contexts.

Findings

Actions are topologically engaging on an open conull dense set.

Results apply to actions preserving unimodular rigid geometric structures of algebraic type.

Supports the use of topological engagement in rigidity theory for Lie group actions.

Abstract

Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a volume density) is topologically engaging on an open conull dense set.