The structure of the embedded submanifold.

Question

If $M$ is a manifold and $S$ is an embedded submanifold, I wanted to understand with which structure we should endow the embedded submanifold. So I was reading wikipedia. The structure they define is not exactly the first that that came to my mind. Intuitively I would have chosen the structure $\{(U\cap S, \phi_{U\cap S}) : (U,\phi) \text{ is a chart for }$M$\}$. Can someone provide me a simple example why this doesn't work?

Answer 1

You cannot (in general) obtain an atlas for a (embedded) submanifold $S \subset M$ by simply restricting an atlas on the ambient manifold $M$. This is because (topological) manifolds are comprised of more than just pure topological data (they are more than just subspaces); they require the data of a topology and (at least) the data of charts. Since charts involve functions with specific properties, they aren't inherited in the same way the open sets are. Recall that an atlas of charts on $S$ is: for every point $p \in S$ an open subset $U \ni p$ equipped with a homeomorphism to an open subset of some $\mathbb{R}^k$.

Of course, the typical counterexample to direct inheritance of charts to topological subspaces is $S = \mathbb{S}^n \subset \mathbb{R}^{n+1} = M$ where $M$ has the single chart $(M,\mathbf{id})$ and $S$ has the subspace topology. Restriction of the identity map $\mathbf{id}:\mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ to $S$ doesn't produce a homeomorphism onto an open subset of $\mathbb{R}^k$ for any $k$.

The obstruction? Restriction of charts isn't sufficiently local (and the nature of manifolds is that they are global objects patched together from local data) because of the homeomorphism to an open set condition.

The fix? Since $S$ is known to be a submanifold, then for each $p \in S$ you should find a chart $(U,\phi)$ from the maximal atlas of $M$ such that $(U \cap S, \phi|_{U \cap S})$ is a chart on $S$. This approach respects the local data on $S$ by importing local, not global, data from $M$. [As it turns out, by the rank theorem you can guarantee that these charts are "slice charts" (cf. the intrinsic structure of a submanifold you see on your wiki link)]

In the example above with $n=1$, if $S$ is the unit circle in $M=\mathbb{R}^2$, and $p=(0,1)$, then you can take the chart $(U,\phi)$ of $M$, where $U = \{y < 1\}$ and $\phi(x,y) = (\frac{x}{1-y}, 1-x^2-y^2)$. Then $(U \cap S, \phi|_{U \cap S})$ is a chart on $S$ (a slice chart, in fact, since it maps to $\mathbb{R} \times {0}$).

The moral? Submanifolds do inherit their structures from the manifold data of the ambient space (as you believed), but you have to dig around inside of $M$'s maximal atlas to restrict an appropriate set of charts. Moreover, if $S$ is any subspace such that every point $p \in S$ is contained in a chart of $M$ that restricts to a slice chart