1 Contact Reduction

For a proper action of a Lie group on a co-oriented contact manifold which preserves the contact structure, we show that the classical Marsden-Weinstein-Meyer symplectic reduction procedure extends naturally to the contact category, continuing the work of V. Guillemin, S. Sternberg, C. Albert, and H. Geiges ([GS1, Al, Ge]). With certain tranasverality and properness assumptions in place, the contact quotient is shown to be naturally a contact manifold. Replacing the transversality condition with one concerning the existence of slices for the co-adjoint action, which is satisfied for compact symmetry groups, the quotient is shown to be a stratified space, extending the results of E. Lerman and the author ([LW]). The strata of the contact quotient are shown to admit natural co-oriented contact structures which vary coherently from stratum to stratum.