Differentiable structure in $S^1$ such that the inclusion map is differentiable

Question

I am asked to explicitly describe a differentiable structure on $S^1$ (with the usual subspace topology) such that the inclusion map $\iota: S^1 \to \mathbb{R}^2$ is differentiable.

I've seen some examples of atlases in $S^1$ but I don't know how to find a maximal atlas such that it defines a differentiable structure on $S^1$ (I don't even understand why the question seem to suggest that there could be more than one differentiable structure on $S^1$) and I don't really understand how would you check that the inclusion map is differentiable.

Answer 1

In order to specify a differential structure, it is sufficient to describe an atlas (I'll come back to why in a second). In order to specify an atlas, it is sufficient to describe a set of charts that

1. cover your manifold,
2. are homeomorphisms onto their images, and
3. are pairwise compatible (transition maps between overlapping charts are differentiable).

To check that a map from a manifold $M$ to $\mathbb{R}^n$ is differentiable, it is sufficient to check that for every chart $U \to M$ in your atlas, the composition $U \to M \to \mathbb{R}^n$ is differentiable in the usual sense.

Finally, returning to differential structures and atlases:

For a given manifold, there are equivalence classes of atlases (where two atlases are equivalent if their union is again an atlas). Each equivalence class defines a differential structure. The identity map $M \to M$ is differentiable with respect to two atlases iff they are in the same equivalence class. In other words, each equivalence class of atlases defines a distinct differentiable structure. For instance, you can look at the number of differentiable structures on spheres $S^m$ for small $m$ here:

https://en.wikipedia.org/wiki/Differential_structure#Differential_structures_on_spheres_of_dimension_1_to_20

In particular, there is only one differentiable structure on $S^1$ as you expect.

The union of all atlases in a given equivalence class is a \emph{maximal atlas}. It therefore doesn't make much sense to describe maximal atlases explicitly, since they will contain many, many charts. Instead, one usually describes a differentiable structure by describing a single atlas, usually with as small a number of charts as possible.