Submanifold In Complex Projective Space Geometry And Topology

In this paper, we mainly study the global and pointwise pinching problems of compact and connected submanifolds in complex projective space.We use the Hopf fibration in the first chapter to study the global pinching problems of minimal submanifolds in complex projective space and we get:Theorem A. Let CP^n+P(4) be a complex projective space of constant holomorphic sectional curvature 4, and φ : M²ⁿ → CP^n+p(4) be a 2n-dimensional real compact minimal submanifold. Denote the scalar curvature of M by R. Ifwhere C(n) is a constant depending only on n, then M is a totally geodesic submanifold CPⁿ.We also discuss the global pinching problems of compact minimal submanifolds in quaternionian projective space, and give out a generalization of the above result.For the general closed submanifolds in complex projective space, by using the Hopf fibration, we prove:Theorem B. Let CP^n+p(4) be a complex projective space of constant holomorphic sectional curvature 4, and φ : M → CP^n+p(4) be a 2n-dimensional compact real Riemannian submanifold without boundary, then there exists a positive constant D(n), depending only on n such thatwhere β_i is the i-th Betti number of M, H is the mean curvature of M.The study of the pointwise pinching problems for Kaehler submanifolds in complex projective space has a long history. Let CP^n+p be an n+p-dimensional complex projective space of constant holomorphic sectional curvature 1, and Mⁿan n-dimensional complete submanifold of CPn+p. K.Ogiue[10] brought forward four famous conjectures in 1970's, the first one and the third one of which have been figured out yet.In the second chapter we study a rigidity problem for Kaehler submanifolds in CPn+p with flat normal bundles, and get the following result:Theorem C. Let Mn{n > 4) be an n-dimensional compact Kaehler sub-manifold of CPn+p with flat normal bundle, denote the sectional curvature of M by K, if K > 0, then Mn is a totally geodesic submanifold CPn.Theorem C. has pushed the study of K.Ogiue's second conjecture.If the holomorphic sectional curvature and the sealer curvature of M satisfies some preconditions, we can have the result as follows:Theorem D. Let Mn be a compact Kaehler hypersurface immersed in CP"+1, if H > S > (1 - n)/2, p > n2 - 45 + 2, then Mn is a totally geodesic hypersurface.