Complete nonnegatively curved spheres and planes

Abstract

We study the space of complete Riemannian metrics of nonnegative curvature on the sphere equipped with C^{k+\alpha} topology. We show the space is homogenous for k>=2. If k is infinite, we show that the space is homeomorphic to the separable Hilbert space. We also prove for finite k, the space minius any compact subset is weakly contractible.