The orientation-preserving diffeomorphism group of \({\mathbb{S}^{2}}\) deforms to SO(3) smoothly (Q649071)

The main result shows that there is a smooth strong deformation retraction \(P:\Omega\times I\to\Omega\) such that \(P_0(f)=f\), \(P_1(f)\in\text{SO}(3)\), \(P_t(A)=A\) and \(P_t(Af)=AP_t(f)\) for each \(t\in I\), \(f\in\Omega\) and \(A\in\text{SO}(3)\). Here \(\Omega\) denotes the orientation-preserving C\(^\infty\) diffeomorphism group of \(\mathbb S^2\) with the C\(^k\) topology. The proof follows the proof of \textit{S. Smale} [Proc. Am. Math. Soc. 10, 621--626 (1959; Zbl 0118.39103)] yielding a non-smooth deformation, but using Sobolev inequalities where necessary to ensure smoothness.