The differentiable sphere theorem for manifolds with positive Ricci curvature

Abstract:

We prove that if $M^n$ is a compact Riemannian $n$-manifold and if $Ric_{\min }>(n-1)\tau _{n}K_{\max }$, where $K_{\max }(x):=\max _{\pi \subset T_{x}M}K(\pi )$, $Ric_{\min }(x):=\min _{u\in U_{x}M}Ric(u)$, $K(\cdot )$ and $Ric(\cdot )$ are the sectional curvature and Ricci curvature of $M$ respectively, and $\tau _{n}=1-\frac {6}{5(n-1)}$, then $M$ is diffeomorphic to a spherical space form. In particular, if $M$ is a compact simply connected manifold with $K\le 1$ and $Ric_M> (n-1)\tau _{n}$, then $M$ is diffeomorphic to the standard $n$-sphere $S^n$. We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension $p$.