Submanifolds Geometric Rigidity Theorem And Differential Sphere Theorem

In this paper, we mainly study several problems on geometry and topology of submanifolds. We obtain rigidity theorems for submanifolds with parallel mean curvature in the unit sphere, differentiable sphere theorems of complete submanifolds, a convergence theorem of yamabe flow on locally conformally flat manifolds, etc. This paper consists of three parts (Chapter 2 to Chapter 4).In Chapter 2, we prove some extrinsic rigidity theorems of submanifolds with parallel mean curvature. In 1986, H. Gauchman[G] proved a famous rigidity the-orem for compact minimal submanifolds in a sphere:Let M be an n-dimensional compact minimal submanifold in Sn+p(1), h be the second fundamental form, ifσ(u)≤1/3 holds for any unit vector u∈UM, where then eitherσ(u)≡0, i.e. M is totally geodesic; orσ(u)≡1/3. Moreover, all minimal immersions satisfyingσ(u)≡1/3 can be determined. In this paper, we generalized Gauchman's theorem as follows:Let M be an n-dimensional complete subman-ifold with parallel mean curvature in the unit sphere Sn+p(1), H be the mean curvature, h be the traceless second fundamental form, if for any unit tangent vector u∈UM, then eitherσ(u)≡0 and M is a totally umbilical sphere; orσ(u)=1/3 and the geometric classification of such submanifold M is given. We also prove a rigidity theorem as follows:Let M be a compact submanifold of unit sphere Sn+p(1) with parallel mean curva-ture, if for any unit tangent vector u,υ∈UM, then M is a totally umbilical sphereIn Chapter 3, we prove differentiable sphere theorems for Riemannian sub-manifolds. The study of curvature and topology of manifolds is one of the core subjects in global differentiable geometry. Andersen, Berger, Brendle, Cheeger, Chern, Colding, Gromoll, Gromov, Grove, Hamilton, Klingenberg, Perelman, Schoen, Shiohama, Yau etc.,have made great contributions to this subject. In this thesis, using the convergence theorem of Ricci flow proved by S. Brendle, we prove the following differentiable sphere theorem:Let M be an n-dimensional complete submanifold of Sn+p(1), andσ(u)<1/3 for any unit tangent vector u∈UM, then Mn is diffeomorphic to Sn(1). We also prove a differentiable sphere theorem for submanifolds with pinched sectional curvature.In Chapter 4, we study the convergence of yamabe flow on locally confor-mally flat manifolds with integral pinching condition. B. Chow [Chow] proved that:Let M be a compact locally conformally flat manifolds with positive Ricci curvature, then the solution of normalized Yamabe flow on M converges in C∞-norm to a constant curvature metric. Recently, H. W. Xu and E. T. Zhao [XZ1] proved that:If M is a locally conformally flat manifolds with constant scalar curvature, and if the Ln/2 norm of traceless Ricci curvature tensor is less than a positive constant, then M is isometric to a space form. Inspired by the results above, we prove the following result:Let M be a compact locally conformally flat manifolds with positive scalar curvature, and if the Lq-norm(q> 2) of the trace-free Ricci curvature is bounded from above by a positive constant, then the solution of the normalized Yamabe flow on M converges to a constant curvature metric. As a consequence, we obtain a differentiable sphere theorem for locally conformally flat manifolds.