Differential Embryo Group And The Moment Map

Atiyah and Bott pointed out that the curvature of a connection on a bundle over a surface can be viewed as the "momentum" corresponding to the action of the gauge group. This observation, together with various extensions, has stimulated a great deal of work and provides a conceptual framework to understand many phenomena in Yang-Mills theory. Our purpose in this paper is to explore similar ideas in the framework of diffeomorphism groups. We hope to show that the moment map point of view is useful , both in understanding certain established results and also in suggesting new problems in geometry and analysis.This paper mainly discuss the topics of whether the moment map exists in the framework of diffeomorphism group (we will see that the moment map exists!) ,and the symplectic quotient in this framework. . At last we study a gradient flow in kahler cases for the purpose of studying the relation between the symplectic and complex quotient, and the role of stability. Firstly, we will see that in the framework of diffeomorphism group the moment map exists and similarly in the framework of symplectic group it exists too. Then we will see that the space of connections of a principal G-bundle over oriented 2-dimensional Riemannian manifold is an infinite-dimensional symplectic manifold , and the action of the gauge transformation on the space of connections is hamiltonian and the moment map is the curvature. And then we will see that in the framework of diffeomorphism group the symplectic quotient is torus bundle over the moduli space of special submanifold. At last we will discuss thegradient flow in (M, ). While the gradient flow equation is not usually parabolic andwe cannot be sure whether solutions exist even for a short time with smooth initial data, so we will examine some sensible cases.