Can Tangent Sphere Bundles over Riemannian Manifolds have Strictly Positive Curvature?

Abstract

We prove, except some particular cases, that for every point x of a Riemannian manifold (M,g), dim M > 2, there is a curvature operator R(X,Y)(X,Y linearly independent) with nontrivial kernel. Then we apply our results to the problem in title.