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 CP 2 (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.

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

“…e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1).…”

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

“…c+|H| 2 with constant holomorphic sectional curvature 4 3 (1 + |H| 2 ). Gu-Xu [17] also obtain the following topological sphere theorem without the assumption of parallel mean curvature vector.…”

“…Theorem B (Ejiri [1] , Gu-Xu [17] ). Let M be an n(≥ 3)-dimensional complete submanifold with parallel mean curvature vector H in F n+m (c) with c + |H| 2 > 0.…”

Geometric Inequalities of Warped Product Submanifolds and Their Applications

“…, m − 1, or the Veronese surface in S 4 . Later on, some new results for the non-existence of the stable currents, vanishing homology groups, topological and differential theorems are well known (see [15][16][17][18][19][20][21][22][23] and references therein). Therefore, it was an objective for mathematicians to understand geometric function theory and topological invariant of Riemannian submanifolds as well as in Riemannian space forms.…”