Applications of diffeological spaces to ordinary differential geometry

Question

Recently I've been learning more about differential geometry, and I came upon the notion of a diffeological space, which encompasses a number of already known extensions of smooth manifolds or related notions, like Banach and Frechét manifolds, complex and analytic manifolds, but also includes a number of other constructions (like quotients and mapping spaces), making the category of diffeological spaces quite well-behaved and nice to work with.

However, I couldn't find much information about applications of diffeology to "ordinary" differential geometry, and would love to hear about some results in this vein. Have diffeological spaces been used to obtain meaningful results about ordinary manifolds (smooth, complex, analytic, p-adic, etc.), specially for cases in which there's no known proof that does not use diffeology?

One example would be something like using the de Rham cohomology of (or other constructions involving) the diffeological space of diffeomorphisms/symplectomorphisms/smooth maps to prove results about ordinary manifolds.

Answer 1

As far as I am aware, you can find some applications of diffeology "merely" in differential geometry of manifolds in the following list (I am not sure this list is exhaustive):

• The (internal) tangent space of the diffeomorphism group of a compact manifold is the space of its vector fields, See Hector G. Géométrie et topologie des espaces difféologiques. Analysis and geometry in foliated manifolds (Santiago de Compostela, 1994). 1995 Nov 17:55-80.or [J.D. Christensen, E. Wu, Tangent spaces and tangent bundles for diffeological spaces, Cah. Topol. G'{e}om. Diff'{e}r. Cat'{e}g. 57(1) (2016), 3-50.].

• The diffeomorphism group of a Lie foliation, See Hector G, Macías-Virgós E, Sotelo-Armesto A. The diffeomorphism group of a Lie foliation. InAnnales de l'Institut Fourier 2011 (Vol. 61, No. 1, pp. 365-378).

• De Rham cohomology of diffeological spaces and foliations, See Hector G, Macías-Virgós E, Sanmartín-Carbón E. De Rham cohomology of diffeological spaces and foliations. Indagationes Mathematicae. 2011 Aug 1;21(3-4):212-20.

• The basic de Rham complex of a singular foliation, See Miyamoto D. The Basic de Rham Complex of a Singular Foliation. International Mathematics Research Notices.

• Basic forms and orbit spaces: a diffeological approach, See Karshon Y, Watts J. Basic forms and orbit spaces: a diffeological approach. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 2016 Mar 8;12:026.

• The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly, See Li J, Watts JA. The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly. Transformation Groups. 2011 Jun;16(2):537-53.

• Smooth Lie group actions are parametrized diffeological subgroups, See Iglesias-Zemmour P, Karshon Y. Smooth Lie group actions are parametrized diffeological subgroups. Proceedings of the American Mathematical Society. 2012 Feb;140(2):731-9.

• Differential forms on manifolds with boundary and corners See Gürer S, Iglesias-Zemmour P. Differential forms on manifolds with boundary and corners. Indagationes Mathematicae. 2019 Sep 1;30(5):920-9.

• The Geodesics of the 2-Torus See here.

• Every symplectic manifold is a (linear) coadjoint orbit See Donato P, Iglesias-Zemmour P. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin. 2022 Jun;65(2):345-60.

• De Rham cohomology for the complement of a (dense) irrational flow on the torus See this MO post and Patrick I-Z's answer

I also believe that even if one is interested in smooth manifolds, it is much easier and conceptual (or perhaps natural) to define "smooth" manifolds through diffeological spaces, just like the definition of topological manifolds. Actually, we can say that a smooth manifold is a diffeological space that is locally Euclidean: every point in the space has an open neighborhood (with respect to the D-topology) which is diffeomorphic to an open subset of a fixed Euclidean space. After that, one can add Hausdorfness and second-countability requirements, if needed. In this situation, there is no need to talk about the smooth compatibility of charts, because it is automatically verified. Notice that diffeological spaces are an extension of Euclidean spaces in the first place.

Answer 2

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X'$ to $X$. Let $\alpha$ be a closed $k$-form on $X$, then $f_1^*(\alpha)$ is cohomologous to $f_0^*(\alpha)$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x' \in X'$ let:

$$ \phi(x') = [s \mapsto f_s(x')], \ \text{then} \ \phi \in C^\infty\big( X',\mathrm{Paths}(X)\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha$ $$ \phi^*[K d\alpha + d K\alpha] = \phi^*(\hat 1^*(\alpha)) - \phi^*(\hat 0^*(\alpha)). $$ Since $d\alpha = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha) = f_0^*(\alpha) + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha). $$ Therefore, $f_1^*(\mathrm{class}(\alpha)) = f_0^*(\mathrm{class}(\alpha))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

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