Diffeomorphisms preserving Morse-Bott functions

Abstract

Let $f:M\to \mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set ${\Sigma }_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f)=\{h\in \mathcal{D}(M)\mid f\circ h=h\}$ the group of diffeomorphisms of $M$ preserving $f$ and let $\mathcal{D}({\Sigma }_f)$ be the group of diffeomorphisms of ${\Sigma }_f$. We prove that the "restriction to ${\Sigma }_f$" map $\rho :\mathcal{S}(f)\to \mathcal{D}({\Sigma }_f)$, $\rho (h)=h{|}_{{\Sigma }_f}$, is a locally trivial fibration over its image $\rho (\mathcal{S}(f))$.

18 pages, the paper is published and we only updated the metadata

References in corpus (3)

• Homotopy types of stabilizers and orbits of Morse functions on surfaces
• Complete connections on fiber bundles
• Symplectomorphisms of surfaces preserving a smooth function, I

Cited by in corpus (3)

• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
• Vector bundle automorphisms preserving Morse-Bott foliations