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 $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi\in \mathrm {Diff}(M)$ preserving $\nabla^gf$ there exists some $t\in\mathbb R$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma,\Phi\_t^g(x)$, where $\Phi\_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.