Research on Poisson manifolds and related structures on manifolds.

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In the second year of the term of the project, we continued to investigate the prop- erties of the plane field which a 2-vector pi defines. A typical example of such plane field is the tangentially symplectic foliation associated with a Poisson structure. In the course of investigation, we obtained the following new insight. (1) In the first year, we showed ; when dim M = 2kappa + 1, if [pi, pi^<kappa>]<double plus> 0, pi defines a contact plane field. This time we have proved the existence of a connection of M such that Divpi (w.r.t. this connection) is a Reel) vector field of the contact structure. (2) If pi defines a (regular) Poisson sturcture, Divpi is a Poisson 1-cocycle and is the image of a modular class (= h_1 in the usual notation of characteristic classes) of the associated folation.This result suggest the possibility of the definition of characteristic classes of singular foliations in terms of Poisson structures in some cases. As a first attempt in this direction, we picked up the left in variant Poisson structures of Lie groups and tried. to descripe the 3-dimensional leaf invarinat (h_3). In a. different direction of singular objects, we studied Dirac structure on manifoIds. A little more time is needed for us to unite these studies arid obtain adenite results.