Singular Reduction and the Stratification Theorem

Abstract

This chapter studies the structure of the symplectic reduced spaces introduced in Chapter 6 when the hypothesis on the freeness of the canonical group action is dropped. In this new scenario, standard momentum maps are not submersions anymore and consequently, the reduced spaces are not necessarily smooth manifolds, but just quotient topological spaces. The main result proved here shows that these quotients are symplectic Whitney stratified spaces in the sense that the strata are symplectic manifolds in a very natural way; moreover, the local properties of this Whitney stratification make it into a cone space in the sense of Definition 1.7.3. This statement is referred to as the Symplectic Stratification Theorem. This symplectic stratification is well adapted to the study of G-invariant dynamics since the flows of Hamiltonian vector fields associated to G-invariant Hamiltonian functions naturally reduce to Hamiltonian systems on these strata.