Geometric, topological and differentiable rigidity of submanifolds in space forms

Abstract

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p(c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n − 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$. In particular, if Ric _M > (n − 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ(n ≥ 4) is a compact submanifold in F ^n+p(c) with c ≥ 0, and if Ric _M > (n − 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.