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MiS Preprint
61/2000
Exotic spheres and the smooth rigidity of the maximal diameter sphere theorem for manifolds of positive Ricci curvature
Wilderich Tuschmann
Abstract
The smooth diameter sphere theorem presented in this note shows that it is possible to isolate the standard sphere among all other complete Riemannian manifolds with positive Ricci curvature by using merely curvature and diameter assumptions, and that in fact any violation of smooth rigidity in Cheng\'s maximal diameter theorem must be accompanied by a blow-up of sectional curvatures: For any given m and C there exists a positive constant $\epsilon=\epsilon (m,C) > 0$ such that any m-dimensional complete Riemannian manifold with Ricci curvature $Ricc \geq m-1$, sectional curvature $K\leq C$ and diameter $\geq \pi - \epsilon$ is diffeomorphic to the standard m-sphere.
