Topological Spere Theorems For Submanifolds With Positive Curvature

In this paper,we obtain some sphere theorems for submanifolds in the space form F^n+p(c). When the K_M is greater than a positive geometric invariant depending on c, n and the mean curvature H, then there is no stable q-currents in M, Moreover M is homeomorphic to a sphere.We also get other sphere theorems.In Chapter 3,we prove:THEOREM A: Let Mⁿ be a compact,oriented n-dimensional submanifold in F^n+p(c) with c≥0.If the sectional curvature of M satisfies:then(1)M is homeomorphic to a sphere if n≠3;(2)M is diffeomorphic to a spherical space form if n = 3.THEOREM B: Let Mⁿ be a compact,oriented n-dimensional submanifold inF^n+p(c) with c > 0.If the sectional curvature of M satisfies:then(1)M is homeomorphic to a sphere if n≠3;(2)M is diffeomorphic to a spherical space form if n = 3.In this paper,we also get a homology vanishing theorem:THEOREM C: Let Mⁿ be a compact,oriented n(n≥4)-dimensional submanifold in F^n+p(c),c + H² > 0, denote by H the mean curvature of Mⁿ.If the sectional curvature of M satisfies:for all 0 < m < n, then H_q(M: Z) = 0, q∈{m,…,n- m}.In Chapter 4,we mainly prove the topological sphere theorem for submanifolds with constant scalar curvature.We get the following: THEOREM D: Let M be an n(n≥3)-dimensional oriented complete submanifoldwith constant scalar curvatureρ=n(n -1)t. If t >n-2/n-1 c,then M is compact,and its fundamental group is finite.Moreover,(1)M is homeomorphic to a sphere if n≥4;(2)M is diffeomorphic to a spherical space form if n = 3.