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 $C^\infty$-diffeomorphismonto$\mathbb{R}^n$ 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.