Diffeologies, Differential Spaces, and Symplectic Geometry

Abstract

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 spaces that do not fall into this category.We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well.We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbittype stratifications induced by a Hamiltonian group action.

Ambrose–singer Theorem on Diffeological Bundles and Complete Integrability of the Kp Equation

“…section 1.4. This is why we felt the need in [14] to compare it to the notion of Frölicher space [4], which appears to complete the first one (this idea has been more recently developped in [23], in a more theoretical approach). This framework seems to us very useful to study infinite systems of equations, and we show that it enables one to use fully justified differential geometric tools ( here, an Ambrose-Singer theorem) in the context of the Kadomtsev-Petviashvili hierarchy already discussed by Mulase [16,18] in a more algebraic way.…”

“…Connection forms are the infinitesimal aspect of G−invariant path-liftings. This leads us to a classification of diffeological structure of a Frölicher space which is less subtle that the one described in [23] and initiated by Iglesias [8] (but sufficient for our purpose), a Lie theorem for Frölicher Lie subalgebras (Theorem 1.18) and the definition of regular Frölicher Lie groups with regular Lie algebra which is directly derived from on the approach of Leslie [12] for analogous notions on diffeological groups. These notions are applied to very easy examples: generalized Lie groups defined by Omori in [19] (Proposition 1.21), and "graded" Frölicher algebras A that appear to be enlargeable into regular Frölicher Lie groups (Theorem 1 .22).…”

The group of diffeomorphisms of a non-compact manifold is not regular

“…A map f : X → X ′ is smooth in the sense of Frölicher if and only if it is smooth for the underlying diffeologies P ∞ (F ) and P ∞ (F ′ ). Thus, we can also state: smooth manifold ⇒ Frölicher space ⇒ Diffeological space A deeper analysis of these implications has been given in [21]. The next remark is inspired on this work and on [13]; it is based on [10, p.26, Boman's theorem].…”

“…[8]. Several definitions and settings have been given by various authors, and the choice that we make to use diffeological spaces as a maximal category, and Frölicher spaces as an intermediate category for differential geometry [13,21], is still quite controversial but this setting becomes developped enough to raise applications. Moreover, historically, diffeological spaces have been developped by Souriau in the 80's with the motivation to deal with the precise objects of interest here, that is groups of diffeomorphisms on non compact, locally compact, smooth manifolds without boundary.…”

“…For basics on this setting, due to Frölicher and Kriegl, see [5,10] and e.g. [13,14,21] for a short exposition on Frölicher spaces and Frölicher Lie groups.…”