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Topological and differentiable rigidity of submanifolds in space forms (1111.2099v1)
Published 9 Nov 2011 in math.DG
Abstract: Let $F{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $Mn(n\geq4)$ is a compact submanifold in $F{n+p}(c)$, and if $Ric_M>(n-2)(c+H2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinching condition above is sharp. Moreover, we obtain a new differentiable sphere theorem for submanifolds with positive Ricci curvature.