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The principle of general covariance has been explicitly described by J.-M. Souriau in his 1974 paper "Modèle de particule à spin dans le champ électromagnétique et gravitationnel" http://www.jmsouria …

There is another way to develop differential geometry without atlases, and even without charts, that is Diffeology. I'm not sure this is the right answer to your question but it worths looking at. Co …

A manifold with corner is a diffeological space modeled on orthants, as such it has a very well defined De Rham cohomology. Edit : With Serap Gürer, we just wrote a paper (will appear in Indag. Math. …

This is a partial answer to what I understand from the question: "...for n=1 we have the "Albanese" map from M into a circle, given by the integration of this form. Can we do something similar for n>1 …

You can look at principal fiber bundles as "half" of groupoids. And for a groupoid right and left actions have a more balanced and obvious meaning. Consider a connected groupoid ${\bf K}$ (that is, b …

Edit: Nov. 26, 2015 Another example about how diffeology represents the smooth structure of orbifolds: the Seifert Orbifolds, as space of fibers of a "Seifert fibered manifold": http://math.huji.ac. …

An addendum to what wrote Peter Michor above: Frölicher spaces identify naturally with the full subcategory of reflexive diffeological spaces. (exercises 79 and 80 of the book). So, for your space equ …

The torus $T^2$ is the quotient of ${\mathbf R}^2$ by ${\mathbf Z}^2$. I denote by $z = [x,y]$ the class of $(x,y) \in {\mathbf R}^2$. I am used to denote $\Delta_\alpha \subset T^2$ what you denote b …

I elaborate a little bit on what your question inspires me. I will treat a more general question, but you can reduce it to finite linear dimensional fiber bundles over manifolds. Let $\pi : E \to X$ a …

If you mean the "Poincaré groupoid" made by fixed-end homotopy classes of smooth paths of a diffeological space, this is a honest diffeological groupoid. Its construction is contained in Chapter V of …

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 …

If you are an algebraist: general sheaves. If you are a geometer: diffeology. I will try to explain what I mean. I may however still modify the following. The philosophy of the theory of sheaves is s …

The first attempt to "quantize" a dynamical variable $u$ on a symplectic manifold $(M,\omega)$, that is, to associate a linear operator $\hat u$ on the space of square summable smooth function $\psi …

I want first to fix a confusion at the level of the two bundles involved in this construction. My way(*) is to consider, first of all, a $S^1$-bundle over a symplectic manifold $(M,\omega)$, with curv …