Annals of Global Analysis and Geometry 14: 427-441, 1996. 427 © 1996 Kluwer Academic Publishers. Printed in the Netherlands. Invariants of Contact Structures and Transversally Ori- ented Foliations AUGUSTIN BANYAGA* Abstract: We exhibit new invariants of the contact structure E(a), the contact flow F, and the transverse symplectic geometry of a contact manifold (M, a). The invariant of contact structures generalizes to transversally oriented foliations. We focus on the particular cases of orientations of smooth manifolds and transverse orientations of foliations. We define the transverse Calabi invariants and determine their kernels. Key words: Contact structures, contact flows, characteristic foliations, simple foliations, transverse symplectic geometry, transverse Calabi and Thurston invariants, affine representations, orientation class MSC 1991: 53C 12, 53C15 1. Introduction A contact form on a smooth manifold M of dimension 2n + 1 3 is a 1-form a such that a A (da)i is everywhere non zero. The contact structure determined by a is the hyperplane field E(a) C TM of kernels of a : Vx E M, E(cr) = {X E TM; a(x)(X) = O}. If the Reeb field of a is complete, it determines a 1-dimensional foliation F,, called the contact foliation. The restriction of the 2-form da to E is a symplectic structure. We call the trans- verse symplectic geometry of the contact manifold (M, a), the geometry of (M, da) which is invariant along Y,. The goal of this paper is to exhibit and study new invariants of the contact struc- ture E(a), the contact flow Fa, and the transverse symplectic geometry (M, da). The invariant for contact structures generalizes to all structures defined by "reg- ular" forms, for instance transversally oriented foliations. We will focus on orienta- tions of smooth manifolds, flows, and transverse orientations of transversally oriented foliations. The invariant for the transverse geometry is a generalization of the Calabi invariant in Symplectic Geometry [2], [14]; we analyse the structure of its kernels. Some results contained in this paper have been announced in [1]. Acknowledgement. This work grew up from the results of the preprint "Au- tomorphisms of flows" I wrote while enjoying the hospitality and support of the * Supported in part by NSF grants DMS 90-01861 and DMS 94-03196.