# Automorphisms of generic gradient vector fields with prescribed finite symmetries

## TL;DR

This paper investigates the automorphisms of gradient vector fields on manifolds with finite group symmetries, showing that such automorphisms are essentially generated by the flow and group actions for generic metrics.

## Contribution

It proves that for generic invariant metrics, automorphisms of gradient vector fields are generated by the flow and the finite symmetry group, clarifying the structure of these automorphisms.

## Key findings

- Automorphisms are generated by the flow and group actions for generic metrics.
- Residual subset of metrics ensures the automorphism structure.
- Automorphisms correspond to elements of the symmetry group and flow time.

## 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 ${\mathcal Met}_f$ with the following property. Let $g\in{\mathcal Met}_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi$ of $M$ preserving $\nabla^gf$ there exists some real number $t$ 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 $\nabla^gf$.