Research On Several Problems Of Kaehler Submanifolds And Totally Real Submanifolds In Complex Projective Space

This dissertatian is mainly concernd with several problems of Kaehler submanifolds and totally real submanifolds in complex projective space. The whole thesis contains two chapters. The main rerults of the thesis are as follows:1. We obtained a necessary condition by generalizating the sufficient and necessary condition for the submanifolds with flat normal bundle in the space of constant cutvature of real manifolds; Obtained a rigidity theorem by using the nesessary condition for the Kaehler submanifolds with flat normol bundle in CP^n+p(1) ; Obtained a better rigidity theorem by adding the isotropic second fundamental condition to the submanifolds.2. We obtained a pinching theorem concerned about scalar curvature for concir-cularly flat Kaehler submanifolds with nonegative Euler number; Otained a boundary estimate theorem concered about scalar curvature for harmonic flat Kaehler submanifolds under the same condition.3. We obtained a Simons-form integral integral inequality and a rigidity theorem for totally real minimal submanifolds. Here we generalized the codimension of the resent literature.4. We obtained a rigidity theorem for totally real minimal submanifolds in CP^n+p(4) with isotropic second fundamental; obtained two theorems about the revelations between A-isotropic totally real submanifolds and submanifolds with isotropic second fundamental; Obtained a integral inequality for A-isotropic submanifolds in CP^n+p(4) with parallel mean curvature vector.5. We obtained a integral inequality and a rigidity theorem for 2-harmonic toally real submanifolds in CPⁿ(4) ; Obtained a rigidity theorem for 2-harmonic pseudo-umibilical submanifolds in CPⁿ(4) .