A Class Of Submanifolds Of Rigid Geometric And Topological Sphere Theorem

In this paper, we mainly study the rigidity of submanifolds with parallel second fundamental form and the curvature and topology of submanifolds in a higher dimensional convex hypersurface.In the first part, we mainly study the hypersurface with parallel second fundamentalform in a pinched Riemannian manifold. We give two geometric inequalities about the square norm of the second fundamental form S:Let Mⁿ be a complete submanifold with parallel second fundamental form immersed in a pinched Riemannian manifold Nⁿ⁺¹ . Denoted by c and d the infinum and the suprenum of the sectional curvature of Nⁿ⁺¹ respectively. Then we haveOur first inequality is better than the one in [10].In the second part, we study the submanifolds with parallel second fundamentalform in space form F^n+p(c)(c≥0), and get the following:Let Mⁿ be a complete connected submanifold with parallel second fundamental form isometrically immersed in a Riemannian manifold with non-negative constant curvature c. Then we haveMoreover, if the equality holds, then M = S^q₁×S^q₂×…S^q_r ((?) (q_i) = n), where S^q_j isthe q_j-dimensional totally umbilical submanifold.In the third part, we study the curvature and topology of submanifolds in a higher dimensional convex hypersurface, and prove the following theorems:Let N^n+p be a hypersurface in R^n+p+1 with its principal curvature satisfy0≤(?)A <1 (A-1,2,…n+p),Mⁿ be an n-dimentional compact submanifold.Denote by H and S the mean curvature and the square norm of the second fundamental form of M respectively. Then where C_n is a positive constant depending only on n,β_i is the i-th Betti numberof M.If (?)[S-nH²+n(n-1)(1-δ)]^n/2dMn, M and Sⁿ are homeomorphous.In general, if 0≤(?) A < (?)(A= 1,2,…n+p) , we have the following results:where C_n is an positive constant depending only on n,β_i is the i-th Betti numberof M.If (?)[S-nH²+n(n-1)(Δ-δ)]^n/2d Mn, Mand Sⁿ are homeomorphous.