The Structure Of The Dirac And Dirac Manifold

Dirac structure on Lie bialgebroid is the subbundle which is maximally isotropic for the natural symmetric fonn ( , )? and whose section is closed under the bracket [,] . Dirac manifold is their important and useful special case.In this paper, we introduce the definition of Lie bialgebroid and its Dirac structure firstly , especially for the definition of characteristic pair of Dirac structure . Moreover , we use directly the conclusions and theorems given by A.,Weinstin, Z-J.,Liu , etc. Based on these , we define the dual characteristic pair of Dirac structure , and get the similar conclusions and theorems , i.e., the if and only if condition of the condition when a maximally isotropic subbundle is a Dirac structure . At the same time , we give the detailed proof, and introduce some special examples . Moreover, we discuss the constrained Dirac structure . These are the general contents of section one.In section two , we view Dirac manifold as the special case of Lie bialgebroid , and get the definition of its characteristic pair and dual characteristic pair .Then , we show some special and useful properties of Dirac manifold. Moreover, we study the submanifold of Dirac manifold and the induced Poisson structure on it in detail. These are the necessary preparation for the later discussion.With the notion of characteristic pair of Dirac structure and the relative theorems , the discussion of Poisson reduction seems to more direct and easier to understand . Moreover , its geometric meaning is more clear . Studying the Poisson reduction from the point of view of Dirac structure was firstly developed by T.J.,Courant . While the introduce and study of characteristic pair was started from Z-J.,Liu , etc. In section three , we make a systematic organization of these contents and give some appropriate adds . The main conclusion and theorem is There is a one-one correspondence between reducible Dirac structures in the double E=TP e T*P and Poisson structures on the quotient space P/F, in which P is a Poisson manifold.Since the particularity of symplectic reduction , there are some difficulties in studying presymplectic reduction . While in particular cases , we can also get some good results , i.e.,under a certain condition there is a one-one correspondence between reducible Dirac structures defined by presymplectic forms and symplectic forms on the quotient spaces . See section four for detail...