If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential...

We develop the theory of smooth principal bundles for a group G, using framework diffeological spaces. After giving new examples showing why arbitrary cannot be classified, we define D-numerable bundles, analogs numerable from topology, and prove that pulling back bundle along smoothly homotopic maps gives isomorphic pullbacks. then structures on Milnor’s spaces EG BG, show → BG is bundle, it c...

In this paper we utilize the notion of infinite dimensional diffeological Lie groups and diffeological Lie algebras to construct a Lie group structure on the space of smooth paths into a completion of a generalized Kac-Moody Lie algebra associated to a symmetrized generalized Cartan matrix. We then identify a large normal subgroup of this group of paths such that the quotient group has the soug...

Abstract We define notions of differentiability for maps from and to the space persistence barcodes. Inspired by theory diffeological spaces, proposed framework uses lifts ordered barcodes, which derivatives can be computed. The two derived (respectively, barcodes) combine together naturally produce a chain rule that enables use gradient descent objective functions factoring through illustrate ...

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus T, n ≥ 2, namely codimension one foliations, flows, and the so-called nonquadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than ±id and translations. The computation is an app...

Abstract The differential-geometric structure of the manifold smooth shapes is applied to theory shape optimization problems. In particular, a Riemannian gradient with respect first Sobolev metric and Steklov–Poincaré are defined. Moreover, covariant derivative associated deduced in this paper. explicit expression leads definition Hessian metric. paper, we give brief overview various techniques...