(12011)GEOMETRIC, TOPOLOGICAL AND DIFFERENTIABLE RIGIDITY OF SUBMANIFOLDS IN SPACE FORMS

来源:数学科学研究中心

Abstract
Let M be an n-dimensional submanifold in the simply connected space form Fn+p(c) with c + H2 > 0, where H is the mean curvature of M. We verify that if Mn(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 CP2( 4 3 (c+H2))
in S7(p 1 c+H2 ). In particular, if RicM > (n − 2)(c + H2), then M is a totally umbilic sphere. We then prove that if Mn(n ≥ 4) is a compact submanifold in Fn+p(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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