Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes

Abstract

There has been much interest among differential geometers in finding relationships between curvature and topology of Riemannian manifolds. For the most part, efforts have been directed towards understanding the implications of positive or negative sectional curvature, Ricci curvature, or scalar curvature, but there are other hypotheses on the curvature which also deserve investigation. In this article, we will consider the topological implications of a new curvature assumption, positive curvature on totally isotropic two-planes, and we will prove via the Sacks-Uhlenbeck theory of minimal two-spheres that a compact simply connected Riemannian manifold of dimension at least four, with positive curvature on totally isotropic two-planes is homeomorphic to a sphere. As corollaries, we will obtain a proof via minimal two-spheres (for Riemannian manifolds of dimension at least four) of the Berger-Klingenberg-Rauch-Toponogov sphere theorem, strengthened to pointwise quarter-pinching, as well as a proof that every simply connected compact Riemannian manifold with positive curvature operators is homeomorphic to a sphere. Let M be an n-dimensional Riemannian manifold with tangent space TpM at the point p E M. Recall that the curvature operator at p is the self-adjoint