Submanifolds Geometry And Topology Study

In this thesis, we mainly study several problems on the geometry and topol-ogy of submanifolds, which contains differentiable sphere theorems for Rieman-nian submanifolds under the conditions of the Ricci curvature, a new pinching theorem for closed hypersurfaces in Sn+1 with constant scalar curvature and constant mean curvature, a geometric characterization of Clifford torus, and some curvature integral inequalities of convex hypersurfaces in 5-dimensional Hadamard manifolds.In Chapter 2, we study differentiable sphere theorems for compact Rieman-nian submanifolds under the conditions of the Ricci curvature. Curvature and topology is one of the core subjects of global Riemannian geometry. Andersen, Berger, Brendle, Cheeger, Chern, Colding, Gromoll, Gromov, Grove, Hamilton, Kingenberg, Perelman, Schoen, Shiohama, Yau, etc., have made great contri-butions to this subject. Recently, H. W. Xu, E. T. Zhao and J. R. Gu [62,65] obtained differentiable sphere theorems for complete submanifolds in space forms under the pinching conditions of the scalar curvature. Using Brendle's Ricci flow technique, we prove differentiable sphere theorems for Riemannian submanifolds under the pinching conditions of the Ricci curvature.In Chapter 3, we obtain a gap theorem of scalar curvature for closed hy-persurfaces in Sn+1 with constant scalar curvature and constant mean curvature. The Simons-Lawson-Chern-do Carmo-Kobayashi theorem for compact minimal submanifolds in a sphere is considered as one of the most important accomplish-ments in the field. This result shows that for closed minimal hypersurfaces in Sn+1, the first gap of the squared length of the second fundamental form is (0, n). In recent 30 years, C. K. Peng, C. L. Terng, S. P. Chang, H. C. Yang, Q. M. Cheng, H. W. Xu, S. M. Wei, Y. J. Suh, H. Y. Yang, etc., have carried out further discus-sions in this direction([40,41,50,53,68]). In this chapter, we extend the result due to Suh and Yang. We prove that:If Mn (n≥4) is an n-dimensional closed hypersurface in Sn+1 with constant scalar curvature and constant mean curva- ture (H≠0). Then there exists an explicit positive constant C(n) depending only on n, such that if|H|< C(n) andβ({n, H)≤S≤β(n,H)+3n/7, then S=β(n, H) and M is a isoparametric hypersurface whereIn Chapter 4, we investigate a geometric characterization of Clifford torus in the unit sphere. The Simons-Lawson-Chern-do Carmo-Kobayashi rigidity theo-rem for closed minimal hypersurfaces in a sphere is an optimal pinching theorem on the scalar curvature. Peng-Terng [41] proved an existence theorem on the second pinched interval of the scalar curvature for n-dimensional (n≤5) closed minimal hypersurfaces in a sphere. In 2007, S. M. Wei and H. W. Xu [53] ex-tended the result of Peng-Terng [41] to the case where n=6,7. Using the lemmas and the approaches in [53], Q. Zhang [69] then gave a proof in dimension 8. This remains to be an open problem for n≥9. In this Chapter, we prove that for an n-dimensional (n≥3) closed minimal hypersurface Mn in Sn+1, if the sum of multiplicities of its maximum principal curvature and minimum principal curva-ture is larger than 2 and if n≤S≤n+2/3, then S(?) n and M is an n-dimensional Clifford torusIn Chapter 5, we study some curvature integrals of convex hypersurfaces in 5-dimensional Hadamard manifolds. There is a long standing isoperimet-ric conjecture which states that for a compact domain with smooth boundary in a Hadamard manifold, one has the same isoperimetric inequality as for Eu-clidean space. The two and four dimensional cases of the Hadamard manifold on this conjecture were settled by A. Weil [54] and C. Croke [21], respectively. B. Kleiner [31] gave a subtle proof in dimension three. Prompted by this motive, A. Brobely[4,5,6] has investigated a differential form due to Chern and derived some results of curvature integral for non-positively curved manifolds very re-cently. In this chapter, we make use of a differential form which differs from that of A. Borbely and prove some integral inequalities of Gauss-Kronecker curvature of convex hypersurface in 5-dimensional Hadamard manifolds.