Local freeness in frame bundle prolongations of $C^\infty$ actions

TL;DR

This paper proves that for a smooth action of a Lie group on a manifold with certain fixed point properties, there exists a dense open subset of the frame bundle where the group action has discrete stabilizers.

Contribution

It establishes the existence of a dense open subset in the frame bundle with discrete stabilizers for actions satisfying specific fixed point conditions.

Findings

Existence of a dense open subset with discrete stabilizers

Applicable to actions with fixed point sets of empty interior

Results hold for actions of dimension at least one

Abstract

Let $G$~be a real Lie group and let $G^\circ$ be the identity component of~$G$. Let $G$~act on a $C^\infty$ real manifold~$M$. Assume the action is $C^\infty$. Assume that the fixpoint set of any nontrivial element of~$G^\circ$ has empty interior in~$M$. Let $n:=\dim G$. Assume $n\ge1$. Let $F$ be the frame bundle of~$M$ of order $n-1$. We prove: there exists a $G$-invariant dense open subset~$Q$ of~$F$ such that the $G$-action on $Q$ has discrete stabilizers.