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 ${\mathcal{M}\mathrm{et}}_f$ with the following property. Let $g\in \mathcal{M}{\mathrm{et}}_f$ and let ${\nabla }^{g}f$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\varphi \in \mathrm{Diff}(M)$ preserving ${\nabla }^{g}f$ there exists some $t\in \mathbb{R}$ 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 ${\nabla }^{g}f$.