分类问题或寻找不变量的问题一直是数学研究中的重要问题。本课题的方向是关于几何算子的分类问题研究。 1978年，M.I.Cowen和R.G.Douglas利用曲率对于Hermitian全纯复丛及其诱导的几何算子进行酉分类，同时几何算子的相似分类问题也被提出。2007年,R.G. Douglas又提出了如何利用曲率来对几何算子进行相似分类的公开问题。本课题的主要内容是利用曲率和第二基本形式作为完全相似不变量来研究全纯复丛及几何算子的相似分类问题，并力图建立该类复丛上相应的相似等价意义下的Swan定理。另一方面，齐次算子以及弱齐次算子类是几何算子中的重要算子类，其具体的刻画是及其困难的，以往人们只有通过群表示论等群理论中技巧来得到齐次算子类的具体刻画。作为应用，我们也希望通过我们关于曲率及第二基本形式的相关研究，能够从算子理论的角度来直接刻画几何算子中的齐次算子以及弱齐次算子类。

Classification or searching the invariant is always an important problem in the researches of Mathematics. In this project, we concern about the researches of the classification of geometry operators. In 1978, by using curvature, M. I. Cowen and R. G. Douglas gave the unitarily classfication of the geometry operators and the complex bundles induced by them. And they also proposed the project of similarity classification of geometry operators. In 2007, R. G. Douglas asked the following open problem : How to use the curvature to give the similarity classification of geometry operators. In this project, we mainly use the curvature and the second fundmental form as invariants to give a similarity classfication of holomorphic bundles and geometry operators and build the Swan theorem in the sense of similarity. On the other hand, the homogeneous operators and weakly homogeneous operators are very important operator class in the Cowen-Douglas operators class. It is very hard to give a clear descripiton of these operators classes. In the past, people only can use the techniques of group representation theory to get a specific characterization of homogeneous operators in Cowen-Douglas classes. As a direct application, we hope there exists a operator theory decripsition of homogeneous operators in geometry operators classes by using our result on the curvatures and the second fundamental forms of the new Cowen-Douglas operator class.