Differential Spaces, Vector Fields, and Orbit-Type Stratifications

TL;DR

This paper explores how Hamiltonian group actions on symplectic manifolds induce natural stratifications on various associated spaces, revealing that these stratifications are intrinsic to the smooth function rings within the differential space framework.

Contribution

It introduces a differential space perspective to analyze stratifications in symplectic geometry, showing their intrinsic nature to smooth functions.

Findings

Stratifications are intrinsic to the ring of smooth functions.

Differential space framework unifies the understanding of stratifications.

Results apply to symplectic quotients and orbit spaces.

Abstract

Let $G$ be a Lie group, and let $(M,\omega)$ be a symplectic manifold. If $G$ admits a Hamiltonian action on $(M,\omega)$ with momentum map $\mu$, then $M$, the zero-level set of $\mu$, the orbit space, and the corresponding symplectic quotient all have induced stratifications. We push this setting into the language of differential spaces, and as a consequence we find that the stratifications are intrinsic to the ring of smooth functions on each space.