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 $\rho(\mathcal{S}(f))$.