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 $\varphi :C^{\infty }(M,G)\to C^{\infty }(M,M)$, defined by $\varphi (\alpha )(x)=\alpha (x)\cdot x$, 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 $\varphi$ 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.

10 pages, this is a translation from russian of the paper published in Proceedings of the conference "Fundamental mathematics today" devoted to the 10-th anniversary of the Independent University of Moscow, 2003, p. 246-258