arXiv research

Transforms Aronszajn to Sikorski subcartesian spaces.

Smooth distributions on subcartesian spaces can be globally finitely generated.

We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.

We discuss properties of the regular part $S_{reg}$ of a subcartesian space $S$. We show that $S_{reg}$ is open and dense in $S$ and the restriction to $S_{reg}$ of the tangent bundle of $S$ is locally trivial.

Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces and almost compl…

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfie…

We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a L…

Study vector fields and flows on singular spaces like submanifolds.

Smooth approximations for continuous functions on orbit spaces.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential…