Automorphisms of generic gradient vector fields with prescribed finite symmetries

Abstract

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal{M}et}_f$ with the following property. Let $g\in {\mathcal{M}et}_f$ and let ${\nabla }^{g}f$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\varphi$ of $M$ preserving ${\nabla }^{g}f$ there exists some real number $t$ and some $\gamma \in \Gamma$ such that for every $x\in M$ we have $\varphi (x)=\gamma {\Phi }_t^g(x)$, where ${\Phi }_t^g$ is the time-$t$ flow of the vector field ${\nabla }^{g}f$.

arXiv comment37 pages. Comments welcome; v2: one reference added, cosmetic changes in the introduction; v3: substantial simplification of the proof following the referee's suggestion, to appear in the Revista Matemática Iberoamericana