Revista Matemática Iberoamericana


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Volume 35, Issue 5, 2019, pp. 1281–1308
DOI: 10.4171/rmi/1083

Published online: 2019-06-07

Automorphisms of generic gradient vector fields with prescribed finite symmetries

Ignasi Mundet i Riera[1]

(1) Universitat de Barcelona, Spain

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 $\nabla^gf$.

Keywords: Vector fields, diffeomorphisms

Mundet i Riera Ignasi: Automorphisms of generic gradient vector fields with prescribed finite symmetries. Rev. Mat. Iberoam. 35 (2019), 1281-1308. doi: 10.4171/rmi/1083