Sections of Lie group actions and a theorem by M. Newman

Abstract

Let $M$ be a smooth finite-dimensional manifold, $G$ be a Lie group, and $\Phi:G \times M \to M$ be a smooth action. Consider the following mapping $\phi: C^{\infty}(M,G) \to C^{\infty}(M,M)$, defined by $\phi(\alpha)(x) = \alpha(x)\cdot x$, for $\alpha\in C^{\infty}(M,G)$ and $x\in M$. In this paper we describe the structure of inverse images of elements of $C^{\infty}(M,M)$ under $\phi$ for $\dim G=1$, i.e. when $G$ is either $\mathbb{R}$ or $S^1$. As an application we obtain a new proof of the well-known theorem by M. Newman concerning the interior of the fixed point set of a Lie group action.