复射影空间中子流形的几何与拓扑研究

【作者】 夏霖；

【导师】 许洪伟；

【作者基本信息】 浙江大学 ， 基础数学， 2006， 硕士

【摘要】 本文主要研究了复射影空间中紧致连通子流形的整pinching问题和逐点pinching问题。在本文第一章中运用Hopf纤维化方法研讨了复射影空间中极小子流形的整体pinching问题,得到了: 定理A.设CP^n+p（4）为具有全纯截面曲率4的复射影空间,φ:M²ⁿ→CP^n+p（4）为CP^n+p（4）中实2n维紧致极小子流形,R为M的数量曲率.如果其中C（n）是仅和n有关的正常数,那么M为全测地子流形CPⁿ。 文中还讨论了四元数射影空间中的紧致极小子流形的整体pinching问题,给出了上述结果的一个推广。 对于复射影空间中一般的闭子流形,本文运用Hopf纤维化方法证明了下述: 定理B.设CP^n+p（4）为具有全纯截面曲率4的复射影空间,φ:M→CP^n+p（4）为CP^n+p（4）中实2n维的紧致黎曼子流形,则存在仅与n有关的正常数D（n）使得其中β_i为M的提升M的第i个Betti数,H为M的平均曲率。 关于复射影空间中Kaehler子流形的逐点pinching问题的研究具有十分悠久的历史。设CP^n+p为具有常全纯截面曲率1的n+p维复射影空间,Mⁿ为CP^n+p中的n维完备Kaehler子流形,K.Ogiue[10]在七十年代曾提出过四个著名猜想。其中的第一和第三个猜想已获解决。 本文在第二章中研讨了CP^n+p中具平坦法丛的Kaehler子流形的一个刚性问题,获得下述结果: 定理C.设Mⁿ（n>4）为CP^n+p中具平坦法丛的n维紧致Kaehler子流形,若M的截曲率K>0,则Mⁿ为全测地子流形CPⁿ。 定理C.推进了K.Ogiue猜想的研究。更多还原

【Abstract】 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.更多还原