Some Notes And Remarks On Holomorphic Bisectional Curvature

People in the complex field research found that when (?)f/(?)zⁱ= 0 , the function f has many interesting things to research. On this basis, if the smooth manifold M^m and the transformation between local coordinates holomorphic, it becomes a complex manifold. When z =x + iy , manifold M can also be seen as it is 2m real dimensional smooth manifold M₀²m and in tangent space T_pM₀, (?)p , an R linear transformation J can be defined such that J²=-id .So {v|Jv = iv} generates holomorphic-tangent space{(?)/(?)zⁱ}where z=1-m , {v|Jv = -iv} generates antiholomorphic tangent space{(?)/(?)zⁱ}and f is known as the complex structure on M .In the study of manifold, Kahler manifold occupies a key position. Kahler pointed out that when the Hermite metric h = h_ijdzⁱ(?)_c dz^j of M meets d(-i/2h_ijdzⁱ∧dz^j) = 0 , its Hermitian connection is the same as Riemann connection. Kahler - Einstein manifold ,i.e. a Kahler manifold with R_ij = cg_ij is a very important manifold.In the research of complex manifold ,holomorphic bisectional curvature is an important tool. T. Frankel, M. Berger, R.L.Bishop, S.I.Goldberg , Kobayashi and so on have come to many conclusions. This paper makes some notes and remarks on the above conclusions so as to make myself comprehension these knowledge completely and provide a good foundation for further study and research in future .In this article , the first part introduces some the basic knowledge of a complex manifolds,for example, some concepts of Kahler manifold and complex vector-bundles . The second part discusses the definition of the holomorphic bisectional curvature, its relationship with Riemann curvature, with Ricci curvature , and its propositions in complex submanifolds The third part discusses an important theorem :An n-dimensional compact connected Kahler manifold with an Einstein metric of positive holomorphic bisectional curvature is globally isometric to P_C^m which has been proved by T. Frankel, Goldberg and Kobayashi in their articles (cf.[3][4]).Also three interesting propositions on complex submanifold of Kahler manifold is mentioned in this article.