Vector Fields and Flows on Subcartesian Spaces

TL;DR

This paper extends the theory of vector fields and flows to singular spaces like submanifolds and quotients, demonstrating that derivations of smooth functions integrate into smooth flows.

Contribution

It develops a framework for integrating derivations into flows on subcartesian spaces, including singular and quotient spaces, within the context of $C^ olinebreak$^ olinebreak{}infty$-ringed spaces.

Findings

Derivations of the $C^ olinebreak$^ olinebreak{}infty$-ring integrate to smooth flows.

Includes analysis of vector fields on singular spaces and quotients.

Provides foundational results for differential geometry on singular spaces.

Abstract

This paper is part of a series of papers on differential geometry of $C^\infty$-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients by actions of compact Lie groups. We show that derivations of the $C^\infty$-ring of global smooth functions integrate to smooth flows.