"Connections in regular Poisson manifolds over R-Lie foliations"

POISSON GEOMETRY, Banach Center Publications, Volume 51, Institute of Mathematics Polish Academy of Sciences, Warszawa 2000.
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The subject of this paper is a notion of the connection in a regular Poisson manifold M, defined as a splitting of the Atiyah sequence of its Lie algebroid. In the case when the characteristic foliation F is an R-Lie foliation the fibre integral operator along the adjoint bundle is used to define the Euler class of the Poisson manifold M. When M is oriented 3-dimensional, a notion of the index of a local flat connection with singularities alng a closed transversal is defined. If additionally F has compact leaves (then F is a fibration over S¹) an analogoue of the Euler-Poincare-Hopf index theorem for flat connections with singularities along closed transversals is obtained.