Geometric, topological and differentiable rigidity of submanifolds in space forms

By Hong-wei Xu, J. Gu. 2011

51 citations recorded by the source corpus.

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 $${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$$CP243(c+H2)inS71c+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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/655ae2cdf23d90b04bf594331b30ec6c78bae9dc