Automorphisms of generic gradient vector fields with prescribed finite symmetries

  • Ignasi Mundet i Riera

    Universitat de Barcelona, Spain
Automorphisms of generic gradient vector fields with prescribed finite symmetries cover
Download PDF

A subscription is required to access this article.

Abstract

Let MM be a compact and connected smooth manifold endowed with a smooth action of a finite group Γ\Gamma, and let ff be a Γ\Gamma-invariant Morse function on MM. We prove that the space of Γ\Gamma-invariant Riemannian metrics on MM contains a residual subset Metf{\mathcal M\mathrm{et}}_f with the following property. Let gMetfg\in\mathcal{M}\mathrm{et}_f and let gf\nabla^gf be the gradient vector field of ff with respect to gg. For any diffeomorphism ϕDiff(M)\phi\in \mathrm {Diff}(M) preserving gf\nabla^gf there exists some tRt\in\mathbb R and some γΓ\gamma\in\Gamma such that for every xMx\in M we have ϕ(x)=γΦtg(x)\phi(x)=\gamma\,\Phi_t^g(x), where Φtg\Phi_t^g is the time-tt flow of the vector field gf\nabla^gf.

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