Geometric, Topological and Differentiable Rigidity of Submanifolds in Space Forms
نویسنده
چکیده
Let M be an n-dimensional submanifold in the simply connected space form F(c) with c + H > 0, where H is the mean curvature of M . We verify that if M(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies RicM ≥ (n−2)(c+H2), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n+1)-sphere with n = even, or CP ( 4 3 (c+H )) in S( 1 √ c+H2 ). In particular, if RicM > (n− 2)(c + H), then M is a totally umbilic sphere. We then prove that if M(n ≥ 4) is a compact submanifold in F(c) with c ≥ 0, and if RicM > (n−2)(c+H2), 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.
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