# Automorphisms of generic gradient vector fields with prescribed finite symmetries

### Ignasi Mundet i Riera

Universitat de Barcelona, Spain

## Abstract

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $Γ$, and let $f$ be a $Γ$-invariant Morse function on $M$. We prove that the space of $Γ$-invariant Riemannian metrics on $M$ contains a residual subset $Met_{f}$ with the following property. Let $g∈Met_{f}$ and let $∇_{g}f$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $ϕ∈Diff(M)$ preserving $∇_{g}f$ there exists some $t∈R$ and some $γ∈Γ$ such that for every $x∈M$ we have $ϕ(x)=γΦ_{t}(x)$, where $Φ_{t}$ is the time-$t$ flow of the vector field $∇_{g}f$.

## Cite this article

Ignasi Mundet i Riera, Automorphisms of generic gradient vector fields with prescribed finite symmetries. Rev. Mat. Iberoam. 35 (2019), no. 5, pp. 1281–1308

DOI 10.4171/RMI/1083