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

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 $n\ge 1$. 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.

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