The space of negative scalar curvature metrics (Q1209141)

Let \(M\) be a closed \(C^ \infty\)-manifold, let \(S^ -(M)\) be the space of Riemannian metrics on \(M\) whose scalar curvature is strictly negative, equipped with the \(C^ \infty\)-Fréchet topology. The main result of the paper is Theorem 1. \(\pi_ i(S^ -(M))=0\) for \(i=0,1,2,\dots\). Combining Theorem 1 with a general result on infinite dimensional topology due to Palais and Whitehead one gets Theorem 2. \(S^ -(M)\) is contractible. Let \(S_{-1}(M)\) be the space of Riemannian metrics whose scalar curvature \(\equiv -1\). After constructing a continuous map \(p:S^ - (M)\to S_{-1}(M)\) with \(p|_{S_{-1}(M)}=id\) one concludes Corollary. \(S_{-1}(M)\) is also contractible.