We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai’s theorem for hyperbolic $3$-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except $S^3$ and $RP^3$, as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a $3$-manifold $X$, the inclusion $operatorname {Isom}(X,g)rightarrow operatorname {Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.
展开▼
