I am bugged about some technical details in the following proof regarding submanifolds. I am reading a book that is not in english , so I'll try to translate here a couple of terms in the best possible way (privileged chart and submanifold chart) but I am not sure thouse are the english names, in case you know them, please tell me.
Definition of submanifold
Let $M$ be an n-manifold and $S$ a subspace of $M$. $S$ is said a submanifold of dimension $d$ of $M$ is for every point $p$ of $M$, there is a chart $(U,x)$ of $M$ such that $x(U\cap S)=x(U)\cap (\mathbb{R}^d\times \{0\}^{n-d})$. In other words, if the points $q$ that fall in $U$ are characterized of the $n-d$ equations $x^{d+1},…,x^{n}(q)=0$. Such a chart is called a privileged chart(with respect to S) and determins a chart $(U \cap S,x_{U \cap S})$, called submanifold chart (if we identify $\mathbb{R}^d$ with$\mathbb{R}^d \times \{0\}^{n-d}$)
If $(U,x), (V,y)$ are privileged charts, we have that $x(U\cap V \cap S)$ is an open set in $x(U\cap S)$ and so in $\mathbb{R}^d$ and $y_{V\cap S} \circ (x_{U\cap S})^{-1}=(y\circ x^{-1})_{x(U \cap V \cap S)}: x(U \cap V \cap S) \to y(U \cap V \cap S)$ is differentiable, so that the submanifold charts form an atlas and determine a d-dimensional differential structure over $S$. With this structure, $S$ is a d-manifold. A d-submanifold is always thought of with this canonical or natural submanifold structure
Proposition: Each chart of the canonical structure of a submanifold S is locally a submanifold chart.
Proof. Let $p$ be a point in the domain of the chart $(V,y)$ of $S$. By definition,…(1) there exists a privileged chart $(U,x)$ of $M$ at $p$ that we can supposed such that $U \cap S \subset V$ and $ x(U)=A \times B$ with $A$ an open set at $0$ in $\mathbb{R}^d$ and B an open set at $0$ in $\mathbb{R}^{n-d}$…(2)
By means of the chart $x$, the canonical projection $\sigma : A\times B \to A\times \{0\}$ induces a sort of projection $\pi = x^{-1} \circ \sigma \circ x :U \to U \cap S$ which is differentiable …(3)
Thus we have that $(U,z)$ with $z=(y^1\circ \pi, …,y^d\circ \pi,x^{d+1},…,x^n)$ with values in $y(U\cap S)\times B$ is a privileged chart and $z|_{U\cap S}=y|_{U\cap S}$…(4)
My questions are:
1 They start using a definition of what?. If it is the definition of a d-submanifold,as I would expect, why aren't they using the definition above, that is $x(U\cap S)=x(U)\cap (\mathbb{R}^d\times \{0\}^{n-d})$)?
2 Why can we suppose $U \cap S \subset V$ and why do we need that? and why do we take opens sets at $0$ necessarily ?)
3 Why is $\pi = x^{-1} \circ \sigma \circ x$ differentiable? My guess is that $\sigma $is differentiable by definition of product topology (if that is applicable here), but still i would need to know that $x $is differentiable for the composition to be differentiable as well, but all I know is that $x$ is a homeomorphism, being $(U,x)$ a chart
4 Why is $z|_{U\cap S}=y|_{U\cap S}$ and why do we need to mention it ?
I'm sorry for all these questions, but I have been around this for days and they are all related to this short proposition,so I don't think it made sense to make independent questions…. can someone please shed some light?
Best Answer
Here they are indeed using the definition of a $d$-manifold, but now requiring as a new assumption the open subset $x(U)$ to be an elementary one of the product space $\mathbb{R}^d\times\mathbb{R}^{n-d}$. We can always assume this point since elementary subsets form a topological basis of the topology of the product space.
The result we want to show is the existence of a privileged chart $(U,x)$ at $p$ with $U\cap S\subset V$ such that $x|_{U\cap S}=y|_{U\cap S}$, so this inclusion is required for the last equality to make sense. Since $V$ is an open set in the subspace topology, then it is by definition of the form $W\cap S$ with $W$ an open set of $M$: take $\widetilde{U}=U\cap W$ if needed. The fact that we choose the chart for the open set $B$ to be at $0$ is important to check the definition you gave of a submanifold, but there are equivalent definitions of being a submanifold where you replace $\mathbb{R}^{d}\times\{0\}^{n-d}$ by $\mathbb{R}^d\times\{q\}$ with $q\in\mathbb{R}^{n-d}$ an arbitrary point (the "difference" is that in this second definition we ask $S$ to locally look like an affine subspace of $\mathbb{R}^n$, meanwhile in the first one we ask it to locally look like a linear subspace of $\mathbb{R}^n$: of course they are equivalent by the mean of translations $x\mapsto x- q$).
When you choose an atlas on a manifold, you are actually making the charts diffeomorphisms between open sets of your manifolds and open sets of $\mathbb{R}^n$ with the usual differentiable structure (just remark that if $x:U\to x(U)\subset\mathbb{R}^n$ is a chart, then $\mathrm{id}_{\mathbb{R^n}}\circ x\circ x^{-1}=\mathrm{id}_{\mathbb{R}^n}$ is a differentiable map between open sets of $\mathbb{R}^n$). Since $\sigma$ is a linear map between open subsets of $\mathbb{R}^n$, then differentiable, and that both $x$ and $x^{-1}$ are diffeomorphisms then differentiable by the above argument, the composition is differentiable.
This is the result we wanted to show, namely the chart of the canonical structure locally being the restriction of a privileged chart. You can first check that it is a chart by finding an explicit inverse, then show that it is a privileged one using the definition, then show that it coincides with $y$ when plotted at points in $U\cap S$ using that $\pi|_{U\cap S}=\mathrm{id}_{U\cap S}$.