| 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 theproperties 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 theexisting 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 . |
