Contact structures: From standard to line bundle approach

Abstract

Motivated by the recent physicists’ interest in contact geometry, this review paper is devoted to some modern geometric insights upon the contact structures. In view of this, we start from the initial perspective on contact manifolds, namely that of an odd-dimensional orientable manifold whose volume form is generated by a 1-form θ and its differential dθ. This naturally arises from a coorientable maximally non-integrable hyperplane distribution. In this picture, we establish a one-to-one correspondence between the transitive Jacobi pairs over odd-dimensional manifolds and the coorientable contact structures over the same manifolds. Then, we introduce the geometric perspective on contact manifolds by omitting the coorientability of the maximally non-integrable hyperplane distribution, and we define the contact structure via an L-valued 1-form [with (L, π, M) a line bundle over an odd-dimensional manifold] with non-degenerate curvature. In this realm, it is shown that there exists a one-to-one correspondence between the transitive Jacobi line bundles over odd-dimensional manifolds and the contact structures over the same manifolds. This faithful ‘representation’ of contact structures brings them nearer to symplectic-like ones through the canonical [bracket] structures inherited from the corresponding Jacobi structures.