In [1] (section 3), C. Scott introduces the following concept of regular atlas for closed $C^\infty$-smooth Riemannian manifolds. He says:
When referring to a coordinate system $(U,\phi)$ as regular, we shall mean that there is another system $(V,\psi)$ with $\overline{U}$ compact, $\overline{U} \subset V$ and $\psi\vert_{U} = \phi$.
His motivation to introduce such concept comes from the problem of defining Sobolev spaces of differential forms. In self-explanatory notation, he defines ($1 \leq l \leq n = \dim M$)
$$\mathscr{W}^{1,p}_{\mathscr{A}}\left( \bigwedge^l M \right) := \left\{ \omega \in \left( \bigwedge^l M \right) : \omega, |\nabla{\omega}| \in L^p \right\},$$
endowed with the norm $\|\omega\|_p + \|\nabla \omega \|_p$. Here,
$$|\nabla \omega(x)|^2 := \sum_{U \in \mathscr{A}} |\nabla_U \omega(x) |^2 = \sum_{U \in \mathscr{A}} \sum_{I,k} \left| \frac{\partial \omega_I}{\partial x^k}(x) \right|^2,$$
where $I = { 1 \leq i_1 < \dots < i_l \leq n}$. Then, he observes
Simple examples demonstrate that it is possible to choose, in perfectly reasonable ways, two atlases which yield Sobolev spaces that are not equivalent as normed linear spaces. […] From here on, classical Sobolev space refers to one constructed as above using a regular atlas. This is all fairly familiar and once again, Morrey [Multiple integrals in the calculus of variations] is a fine reference.
Such observation is not proved nor references to the literature are provided for the counterexamples. Morever, I have not found a similar definition in Morrey's book (Morrey deals with admissible cooordinate systems, see pp. 288 and 300, which are defined differently and it appears that the only requirement about them is the standard compatibility between new and old coordinates). In addition, it seems to me (but I'm quite new to this topic) that this problem is not considered by other authors in this area.
Exceptions are [2] and [3]. In [2] (section 2.2), Iwanienc, Scott and Stroffolini, dealing now with smooth Riemannian manifolds with boundary, give the following definition of regular atlas. If $\mathscr{R}$ is a $C^\infty$-smooth, closed Riemannian manifold,
a regular open region $M \subset \mathscr{R}$ is one for which there exists a finite atlas $\mathscr{A}$ on the reference manifold $\mathscr{R}$ consisting entirely of coordinate charts $(U,\kappa) \in \mathscr{A}$ so that $\kappa$ is a $C^\infty$-diffeomorphism onto $\mathbb{R}^n$ and $\kappa(U \cap M) = \mathbb{R}^n_+$ whenever $U$ meets $\partial M$.
In [3], section 2.1.1, the authors are concerned only with closed manifolds. The requirement is the same but obviously there is no need for the second part (as $\partial M = \varnothing$).
It seems to me that no one of these requirements is common in standard differential geometry. For instance, I guess that the sphere $\mathbb{S}^n$ with the most common atlas consisting of the two coordinate patches associated with the stereographic projections from the north and south poles is not regular in the sense of Scott, and this is perhaps quite weird.
Summarizing, these are my questions (they are multiple but strictly tied):
- Is it always possible to introduce such atlases?
- Are the two definitions in [1] and [3] equivalent?
- Are they necessary to give a meaningful definition of Sobolev spaces of differential forms? What are the easy counterexamples mentioned above?
- Are they necessary to prove the $L^p$-version of Gaffney's inequality and Hodge-de Rham-Kodaira decomposition, which are essentially the main results of [1] and [2]?
[1] C. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (6), 1995.
[2] T. Iwaniec, C. Scott, B. Stroffolini, Nonlinear Hodge theory on manifolds, Ann. Mat. Pura Appl. (IV), CLXXVII (1999), 37-115.
[3] P. Hajlasz, T. Iwaniec, J. Maly, J. Onninen, Weakly differentiable mappings between manifolds, AMS.
Best Answer
Ok this is already quite a mouth full, so let me try to give answers to some of your questions: The main issue is that Sobolev mappings are defined via a boundedness concept (you ask for $L^p$-integrability conditions) and boundedness is not an intrinsic concept on a manifold. This means that definitions in charts usually blow up when boundedness is concerned (as I can always compose with an arbitrary diffeomorphism to make new charts and by choosing badly I can blow up every such condition). This is often glossed over in the older Sobolev literature on manifolds (or they define things chart independent in which case the problem does not exist). For most of what follows now, I refer to the great memoir by Inci, Kappeler and Topalov: On the regularity of the composition of diffeomorphisms (arxiv version is here 1) where these things are worked out.
Atlases with the desired property always exist. Indeed one can even strengthen the conditions to make the atlases more convenient to work with (e.g. Lipschitz boundary so restrictions of your Sobolev maps behave nicely). Atlases with the additional properties are called fine cover in 1 (see Section 3.1 in 1 for discussion and an existence proof)
These atlases are necessary to talk about Sobolev functions on a manifold in charts. There are chart independent definitions which usually require you to pick a Riemannian metric as an additional structure to make sense of integration and boundedness. For example the classical book by Palais: Foundations of Global Non-Linear Analysis, 2 does it this way. See the Appendix of 3 for a comparison of the notion (even in the boundary case).
For the counterexample we can look at 1 p.43 (which I am quoting here): Consider the torus $M= \mathbb{R}/\mathbb{Z}$ and let $f\colon (−1/2, 1/2) \rightarrow \mathbb{R}$ be the function $$f(x) := \begin{cases} x^{2/3} &, x \in [0, 1/2)\\ (−x)^{2/3} &, x \in [−1/2, 0). \end{cases}$$ Extending $f$ periodically to $\mathbb{R}$ we get a function on $M$ that we denote by the same letter. It is not hard to see that $f \in H^1(M, \mathbb{R})$. Now, introduce a new coordinate $y = x^2$ on the open set $(0,1/2) \subseteq M$. Then $\tilde{f}(y) := f(x(y)) = y^{1/3},\ y \in (0, 1/4)$. We have, $\tilde{f}'(y) = 1/(3y^{2/3})$, and hence, $\tilde{f}′ \not\in L^2((0, 1/4), R)$. This shows that $\tilde{f} \not \in H^1((0, 1/4), R)$ and thus provides an example showing that a function may fail to be Sobolev in certain coordinate charts (and the boundedness conditions remedy that). Upshot: Sobolev functiobs are not stable under arbitrary change of charts. Similar ideas will also produce differential forms which are $H^s$ in one but not the other coordinate system.
This answers half of your questions (since I am no expert on Sobolev spaces I will leave the other questions to somebody who is more in the know).