Research On The Pinching Of The Minimal Submanifolds And Kaehler Submanifolds

In this paper,the second basic form of the Laplacian operator is calculated by the active frame method,and the relationship between the second basic form length square and the sub-manifold full-measuring ground is mainly studied.The specific contents include:· In Chapter 1,we introduce the research background and research significance of submanifold geometry,and the research status of submanifold geometry in re-cent years.Through in-depth analysis of the research background and research status,this paper describes the main issues of this study.· In Chapter 2,we give the basic concepts,symbols,and some related lemmas of Riemannian manifolds.In the first part,when the outer space is a Riemannian manifold with constant curvature,the Laplacian operator of the second basic form is calculated,and some conclusions about the complete geodetic survey of minimal submanifolds are given.In the second part,the peripheral space is extended to a quasi-constant curvature Riemannian manifold.The Laplacian operator of the second basic form is calculated using a similar method,and the results of the complete geodesy of minimal submanifolds are generalized· In Chapter 3,we give the basic concepts and symbols of the complex projective space.When the outer space is a constant curvature complex projective space,the Laplacian operator of the second basic form of the Kaehler submanifold with a flat bundle is calculated,and some conclusions are obtained.· In Chapter 4,we summarize the main results in this paper and give some prospects for further research in the future.