Vector Fields and Flows on Subcartesian Spaces

This paper is part of a series of papers on differential geometry of $C^{\mathrm{\infty }}$-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 $C^{\mathrm{\infty }}$-ring of global smooth functions integrate to smooth flows.