A sharp differentiable pinching theorem for submanifolds in space forms

Abstract:

Let $M$ be an $n$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$. We verify that if the sectional curvature of $M$ satisfies $K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $M$ is diffeomorphic to a spherical space form. Moreover, we show that if $M$ is an oriented compact submanifold in $F^{n+p}(c)$ with $c\ge 0$, and if $n\neq 3,5$, $K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $M$ is diffeomorphic to the standard $n$-sphere. It should be emphasized that our results are optimal for $n=4$.