流形上一类新的算子的研究
摘要
本文的选题首先紧扣当前国内外微分几何研究的大趋势、大潮流，并考虑了当前微分几何理论研究的几个具体方面：
1．黎曼流形，包括紧致与非紧致黎曼流形上几何性质与拓扑结构的研究；齐性空间与对称空间的几何性质及其与李群之间的关系；三维欧氏空间曲面的整体性质；子流形特别是极小流形的研究等。
2．流形上各种算子(如偏微分算子)的研究。
3．纤维丛几何，包括纤维丛上的联络论、示性类的研究及其应用等。
4．复流形几何。
5．代数微分几何。
本文选择和尝试对流形上一类新的算子的研究。
全文分五个部分：第一部分介绍新算子Δ的概念，包括在坐标变换下的变换规则、对函数l_j((?))的研究、算子Δ的四条性质，以及在奇异点处算子Δ的性质；第二部分证明了在流形M上算子Δ的存在性，以及Δ算子形式选取的任意性；第三部分对给出的新算子Δ，定义了流形上新的平移概念，由此得到一个新的向量场，对比新旧向量场，可以得到Δ算子的几何解释；众所周知李导数和协变微分在微分几何和现代物理中是最基本、最重要的两类算子，在本文第四部分阐述了李导数和协变微分是Δ算子的特殊形式，这是本文的创新点：本文的最后一部分，尝试将Δ算子应用于切丛，并得到了Δ算子的几何解释。
The choice of this article is firmly tightened with the trend of the present study of the differential geometry at abroad and at home, and considers some concise themes of study on theory of differential geometry. They are as follows:
1. Riemann manifolds.
2. Study of all kinds of operators (such as partial differentia operators) on manifolds.
3. Geometry of fiber bundles.
4. Geometry of complex manifolds.
5. Algebraic differentia geometry, rigorously speaking, which doesn't belong to category of differentia geometry.
The purpose of this paper is to form a new wider operator on the manifold M. This goal will be achieved in five parts: the fust part is an introduction of a operator. We begin with the construction of a new operator on manifolds, including its definition, regulation under coordinate alternate and the study on function lij(x), and discuss its own properties and the
properties at the singular point. In the second part I prove the general existence of such an operator; in the next part assuming that an operator a has been selected in the manifold M, then we introduce a concept of transport of vector field under the definition of a,and gain a new vector field. Comparing the
existing and the getting, we surprisedly obtain the geometrical interpretation of a In part four we will prove that the general framework for a operator reduces to, in special cases, covariant differentiation and Lie differentiation, which are fundamental importance in differentia geometry, and this is an important innovation of this paper. In the last part I apply a operator into tangent bundles, and also obtain the geometrical interpretation of a .
1．黎曼流形，包括紧致与非紧致黎曼流形上几何性质与拓扑结构的研究；齐性空间与对称空间的几何性质及其与李群之间的关系；三维欧氏空间曲面的整体性质；子流形特别是极小流形的研究等。
2．流形上各种算子(如偏微分算子)的研究。
3．纤维丛几何，包括纤维丛上的联络论、示性类的研究及其应用等。
4．复流形几何。
5．代数微分几何。
本文选择和尝试对流形上一类新的算子的研究。
全文分五个部分：第一部分介绍新算子Δ的概念，包括在坐标变换下的变换规则、对函数l_j((?))的研究、算子Δ的四条性质，以及在奇异点处算子Δ的性质；第二部分证明了在流形M上算子Δ的存在性，以及Δ算子形式选取的任意性；第三部分对给出的新算子Δ，定义了流形上新的平移概念，由此得到一个新的向量场，对比新旧向量场，可以得到Δ算子的几何解释；众所周知李导数和协变微分在微分几何和现代物理中是最基本、最重要的两类算子，在本文第四部分阐述了李导数和协变微分是Δ算子的特殊形式，这是本文的创新点：本文的最后一部分，尝试将Δ算子应用于切丛，并得到了Δ算子的几何解释。
The choice of this article is firmly tightened with the trend of the present study of the differential geometry at abroad and at home, and considers some concise themes of study on theory of differential geometry. They are as follows:
1. Riemann manifolds.
2. Study of all kinds of operators (such as partial differentia operators) on manifolds.
3. Geometry of fiber bundles.
4. Geometry of complex manifolds.
5. Algebraic differentia geometry, rigorously speaking, which doesn't belong to category of differentia geometry.
The purpose of this paper is to form a new wider operator on the manifold M. This goal will be achieved in five parts: the fust part is an introduction of a operator. We begin with the construction of a new operator on manifolds, including its definition, regulation under coordinate alternate and the study on function lij(x), and discuss its own properties and the
properties at the singular point. In the second part I prove the general existence of such an operator; in the next part assuming that an operator a has been selected in the manifold M, then we introduce a concept of transport of vector field under the definition of a,and gain a new vector field. Comparing the
existing and the getting, we surprisedly obtain the geometrical interpretation of a In part four we will prove that the general framework for a operator reduces to, in special cases, covariant differentiation and Lie differentiation, which are fundamental importance in differentia geometry, and this is an important innovation of this paper. In the last part I apply a operator into tangent bundles, and also obtain the geometrical interpretation of a .
引文
1 W M Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press. Orlando.
2 Howard Osborn. Vector Bundles. Academic Press. New York.
3 陈维桓．微分流形初步．北京大学出版社
4 陈省身．陈维桓．微分几何讲义．北京大学出版社
5 杨万年．微分流形及其应用．重庆大学出版社．
6 侯伯元．侯伯宇．物理学家用微分几何．北京科学出版社
7 [英]B．F．舒次．数学物理中的几何方法．上海科学技术文献出版社
8 詹汉生．微分流形导引．北京大学出版社
9 伍鸿熙．沈纯理．虞言林．黎曼几何初步．北京大学出版社
10 梅向明．李群讲义．
11 王宝勤．外微分与纤维丛讲义．
12 白正国．沈一兵．水乃翔．郭孝英．黎曼几何初步．高等教育出版社．
2 Howard Osborn. Vector Bundles. Academic Press. New York.
3 陈维桓．微分流形初步．北京大学出版社
4 陈省身．陈维桓．微分几何讲义．北京大学出版社
5 杨万年．微分流形及其应用．重庆大学出版社．
6 侯伯元．侯伯宇．物理学家用微分几何．北京科学出版社
7 [英]B．F．舒次．数学物理中的几何方法．上海科学技术文献出版社
8 詹汉生．微分流形导引．北京大学出版社
9 伍鸿熙．沈纯理．虞言林．黎曼几何初步．北京大学出版社
10 梅向明．李群讲义．
11 王宝勤．外微分与纤维丛讲义．
12 白正国．沈一兵．水乃翔．郭孝英．黎曼几何初步．高等教育出版社．