Finite $C^\infty$-actions are described by one vector field

TL;DR

This paper demonstrates that any finite group action on a smooth manifold can be realized as the automorphism group of a complete vector field, extending understanding of symmetries in differential topology.

Contribution

It shows that for any finite subgroup of diffeomorphisms on a connected smooth manifold, there exists a complete vector field whose automorphism group precisely matches that subgroup times the real line.

Findings

Finite group actions can be realized as automorphism groups of vector fields.

Constructs a complete vector field with prescribed finite symmetry group.

Extends the understanding of symmetry realization in smooth manifolds.

Abstract

In this work one shows that given a connected $C^\infty$-manifold $M$ of dimension $\geq 2$ and a finite subgroup $G\subset \Diff(M)$, there exists a complete vector field $X$ on $M$ such that its automorphism group equals $G\times \mathbb{R}$ where the factor $\mathbb{R}$ comes from the flow of $X$.