Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

A contact manifold is a pair (M, ξ) consisting of an odd-dimensional manifold M endowed with a one-codimensional completely non-integrable distribution ξ on M. If ξ = ker η for a differential one-form η on M, which is not unique, the pair (M, η) is called a co-orientable contact manifold. In this talk, I will present a Marsden–Weinstein reduction for co-oriented contact manifolds devised in C. Willett, Contact reduction, Trans. Amer. Math. Soc. 354 (2002) 4245–4260. Roughly speaking, this reduction uses a Lie group action of symmetries of η to obtain from it a new co-oriented manifold on a quotient of a submanifold of M in a manner that does not depend on the choice of η corresponding to a fixed ξ. More in detail, I will first introduce some basic notions on co-oriented contact manifolds and briefly explain how they can be viewed as symplectic R^×-principal bundles. In particular, I will recall the definition of a contact Lie group action and its associated momentum map. Next, I will introduce the notion of an orbifold and a contact quotient. Finally, I will present a co-oriented contact Marsden–Weinstein reduction theorem that ensures that the contact quotient is a contact orbifold. To understand the more complicated case of a co-oriented contact orbifold, I will examine a symplectic orbifold obtained through symplectic reduction. If time permits, I will discuss removing the singularity and the integrability assumptions in favour of the existence of a slice.