Induced smooth structure on on $S^m$ coincides with the one defined by the stereographic projection

Question

Considering the regular value theorem one can look at the $m$-dimensional sphere $S^m \subset \mathbb{R}^{m+1}$ as an embedded submanifold of euclidean space (as the zero set of a smooth submersion). Immediately after this statement I am asked to verify that the 'induced smooth structure' on $S^m$ coincides with the one defined by the stereographic projections. What I do not quite get, what exactly is this 'induced smooth structure'?

In the 'reminder on multivariable analysis'-part of the lecture notes the equivalent definitions of embedded submanifolds of $\mathbb{R}^m$ are discussed. Including the characterizations:

1) A subset $M \subset \mathbb{R^m}$ admits a d-dimensional chart around $p \in M$ (a diffeomorphism onto an open neighborhood of $\mathbb{R^p}$).

2) $M$ can be described around $p$ by a d-dimensional implicit equation (a submersion).

My understanding of the questions is that I should find charts by making use of the implicit equation (thus finding the explicit charts for the sphere that proves the equivalence 2) $\rightarrow 1)$) and then check that these charts are smoothly compatible with the stereographic projection. Problem is that I have no clue whatsoever as to what these charts should look like. Also, more generally in proving this equivalence, how can one construct charts starting from this implicit description?

Answer 1

The "induced smooth structure" is the one you just defined via the smooth submersion $x_1^2+...+x_n^2$.

So you need to check that defining a smooth structure on $S^m$ via this submersion produces the same thing as defining it via the stereographic projection

Answer 2

You do not need to find the charts explicitly. You can work with the equivalence you just provided. Given the m-dim implicit equation (which is a submersion), and given the equivalence between characterization 1) and 2), you know there are d-dim charts around p in M for every p in M. These charts define an atlas (since they are diffeomorphisms) and thus the induced smooth structure on the m-sphere.

Now al you need to show is that this atlas obtained through the submersion and the equivalence provided by you is smoothly equivalent to the atlas defined by the stereographic projections. (You do not need the charts explicitly, just work with them abstractly). Then it follows that they are in the same equivalence class and thus have the same smooth structure (maximal atlas).