Chapter 3: Submanifolds and Embeddings

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Chapter 3: Submanifolds and Embeddings

Contents
Contents
 1.  Submersions, immersions, embeddings
   1.1.  Local diffeomorphisms
   1.2.  Constant rank theorem
   1.3.  Embeddings
 2.  Submanifolds
   2.1.  Embedded submanifolds
   2.2.  Level sets
   2.3.  The tangent space to a submanifold

In this chapter we develop the theory further with differentials.
1. Submersions, immersions, embeddings We can use the tool of linear algebra to analyze the local structure of a smooth map since \mathrm dF _ p is linear. The most important property of a linear map is the rank. The maps with constant rank are of special interest.

Let M,N be smooth manifolds and F:M\to N be a smooth function, and p\in M.

From the knowledge of linear algebra, the rank of F at p is the rank of the Jacobian of F in any chart, and is the dimension of \operatorname{Im}\mathrm dF _ p.

If the differential \mathrm d _ pF is surjective or injective at a point p, then F is locally a submersion or immersion. Choose any chart containing p. Then since \mathrm d _ pF is surjective or injective, the Jacobian of F in coordinates at p has full rank r, so there is a nonsingular r\times r submatrix of the Jacobian. By continuity of determinant this submatrix is still nonsingular in a neighborhood of p, indicating the Jacobian is still of full rank. We obtain the following proposition.

1.1. Local diffeomorphisms Let F:M\to N be a diffeomorphism. We have known the differential \mathrm dF _ p is an isomorphism. What if we are given the differential \mathrm dF _ p of a map F is an isomorphism? It turns out that F is diffeomorphism locally.

Note that local diffeomorphism does not necessarily imply (global) diffeomorphism. For example the map f:\mathbb R\to S^1 defined by x\mapsto\mathrm e^{2\pi\mathrm ix}. This f is not injective. If we assume F:M\to N is a bijective local diffeomorphism, then it is easy to see it is a global diffeomorphism.

If F is a local diffeomorphism the result is easy to check. Conversely, we need the inverse function theorem for manifolds.

This theorem can be proved by applying the inverse function theorem for Euclidean spaces to \widehat F=\psi\circ F\circ\varphi^{-1} where (U,\varphi) is a chart in M containing p and (V,\psi) is a chart in N containing F(p). Note that \varphi,\psi are diffeomorphisms and \mathrm d\widehat{F}=\mathrm d\psi\circ \mathrm dF\circ\mathrm d(\varphi^{-1}).
1.2. Constant rank theorem The inverse function theorem for manifolds has a very important consequence, the constant rank theorem. By constant rank theorem, a smooth map with constant rank can have a particularly simple form of coordinate representation, if we choose charts properly.

Proof. We first consider the special case in which M= \mathbb R^m, N= \mathbb R^n.

The Jacobian of F at p has rank r, so there is a nonsingular r\times r submatrix. We may assume it is the upper left submatrix, otherwise we reorder the coordinates. Now suppose F=(F^1,F^2) where F^1 is the first r components and F^2 is the last n-r components. We have \det J _ 1F^1(p)\neq0.

Define G:U\to\mathbb R^m.
\begin{gather*}
G(x,y)=(u,v)=(F^1(x,y),y). \\
JG(p)=\begin{bmatrix}
J _ 1F^1(p) & J _ 2F^1(p) \\
0 & I _ {m-r}
\end{bmatrix}.
\end{gather*}Since \det JG(p)=\det F^1(p)\neq0, by inverse function theorem there are neighborhoods U _ 1\subseteq\mathbb R^m and V _ 1\subseteq\mathbb R^m such that G:U _ 1\to V _ 1 is a diffeomorphism. On V _ 1,
\[
(u,v)=(G\circ G^{-1})(u,v)=(F^1\circ G^{-1},y\circ G^{-1})(u,v).
\]Comparing components we have u=F^1\circ G^{-1}(u,v). Therefore if we set Q=F^2\circ G^{-1} then
\[
(F\circ G^{-1})(u,v)=(u,(F^2\circ G^{-1})(u,v))=(u,Q(u,v)).
\]
Note that G^{-1}:V _ 1\to U _ 1 is a diffeomorphism and F has constant rank r on U _ 1. We can see F\circ G^{-1} has constant rank r on V _ 1. We compute the Jacobian:
\[
J(F\circ G^{-1})=\begin{bmatrix}
I _ r & 0 \\
J _ uQ & J _ vQ
\end{bmatrix}
\]The first r columns are independent, so in order to have constant rank r, on V _ 1 the Jacobian J _ vQ must be zero, indicating Q(u,v)=Q(u). Then we can write (F\circ G^{-1})=(u,Q(u)).

Now let H:\mathbb R^m\to\mathbb R^m be H(x,y)=(x,y-Q(x)). Then
\[
(H\circ F\circ G^{-1})(u,v)=F(u,Q(u))=(u,0).
\]Note that G, H are both diffeomorphisms, and it can be checked that we can translate them such that G(p)=0 and H(F(p))=0. The Euclidean version of the theorem is proved.

We return to the general case in which M,N are manifolds. Choose a chart (U,\tilde \varphi) containing p in M and chart (V,\tilde \psi) containing F(p) in N. We apply the theorem for Euclidean spaces to \tilde\psi\circ F\circ\tilde\varphi^{-1}, for it has the same constant rank r as F. Then there is a diffeomorphism G in a neighborhood of \tilde\varphi(p) and a diffeomorphism H of a neighborhood of \tilde\psi(F(p)) such that (H\circ\tilde\psi\circ F\circ\tilde\varphi^{-1}\circ G^{-1})(x^1,x^m)=(x^1,\dots,x^r,0,\dots,0). Set \varphi=G\circ\tilde\varphi and \psi=H\circ\tilde\psi. The proof is done.

.

We investigate this theorem from an another point of view. Let q\in F(U) and P be the projection in the first r coordinates and P _ \perp be the projection in the last n-r coordinates. If q=(y _ 1,y _ 2) then Pq=(y _ 1,0) and P _ \perp q=(0,y _ 2). From the proof we have q=(y _ 1,y _ 2)=(F\circ G^{-1})(u,v)=(u,Q(u)) for some (u,v). Therefore P _ \perp q is determined by Pq, so
\[
q=Pq+P _ \perp q=Pq+R(Pq),\quad q\in F(U).
\]The point q is also determined by projection Pq.
1.3. Embeddings A special kind of immersion is embedding. Recall that F:M\to N is a topological embedding if it is a homeomorphism onto F(M) equipped with the subspace topology of N.

For example, if U\subseteq M is an open submanifold, the inclusion map \iota:U\hookrightarrow M is smooth embedding. There are also some counterexamples. The map \gamma:\mathbb R\to\mathbb R^2 defined by \gamma(t)=(t^3,0) is a smooth map and a topological embedding, but it is not an immersion since at 0 it is not injective: \gamma'(0)=0. Moreover, the map \beta:(-\pi,\pi)\to\mathbb R^2 defined by \beta(t)=(\sin 2t,\sin t) is an injective smooth immersion, but it is not a topological embedding, for the image is compact in subspace topology while the domain is not compact.

How can an injective immersion be an embedding? This is not easy to answer. However a simple criteria is given in the following proposition.

To see it we just need to show f^{-1}:f(M)\to M is continuous. Let A be a closed set in M. Then the compactness of M implies the compactness of A. Since a continuous map maps a compact set to a compact set, we know f(A) is compact. This means that f^{-1} maps a closed set f(A) to a closed set A, indicating f^{-1} is continuous. With this proposition we can prove the following theorem.

If for every p\in M locally F|U is an embedding, then it is clear that F has full rank at M, so F is an immersion. Conversely, suppose F is an immersion. Then p has a neighborhood U such that F| _ U is injective, and further has a precompact (i.e., with compact closure) neighborhood U _ 1 such that \overline{U} _ 1\subseteq U. By the previous proposition we knowF| _ {\overline{U} _ 1} is an embedding. Then F| _ {U _ 1} is also an embedding.
2. Submanifolds In this section we study smooth submanifolds. The most important type is the one called embedded submanifolds.
2.1. Embedded submanifolds Let M be a smooth manifold. We define an embedded manifold as follows.

This theorem characterise embedded submanifold in another way. The proof of it is not covered here. We may assume the coordinate functions vanish except for the first k of them. Then \varphi(S) is of the form
\[
\{(x^1,\dots,x^k,x^{k+1},\dots,x^n)\in \varphi(U)\mid x^{k+1}=\dots=x^n=0\}.
\]

It is clear that with the subspace topology S is a topological manifold. We give S a smooth structure by taking smooth charts to be those of the form (F(U),\varphi\circ F^{-1}) where (U,\varphi) is any smooth chart in N. These charts are smooth compatible since those charts in N are compatible. With this smooth structure F is a diffeomorphism. Since F^{-1} is a diffeomorphism and F is an embedding, the inclusion map \iota:S\hookrightarrow M which is the composition F\circ F^{-1} is an embedding.
2.2. Level sets Let \Phi:M\to N be a map and c\in N. The set \Phi^{-1}(c) is called a level set of \Phi.

A point p\in M is said to be a regular point of \Phi if \mathrm d\Phi _ p is surjective. It is a critical point otherwise. Then F is a submersion iff every point is regular. If every point of \Phi^{-1}(c) is a regular point, then c is called a regular value, and \Phi^{-1}(c) is a regular level set; and a critical value otherwise.

If S is an embedded submanifold of M, then \dim M-\dim S is called the codimension of S in M.

The level set S=\Phi^{-1}(c)\subseteq M is an embedded submanifold in M. Let p\in \Phi^{-1}(c). By the constant rank theorem (Theorem 7) there is a chart (U,\varphi) in M centered at p and a chart (V,\psi) in N centered at \Phi(p) such that
\[
(\psi\circ \Phi\circ\varphi^{-1})(x^1,\dots,x^m)=(x^1,\dots,x^r,0,\dots,0)\in\mathbb R^n.
\]Thus the level set (\psi\circ \Phi\circ\varphi^{-1})^{-1}(0) is defined by the vanishing of the coordinates x^1,\dots,x^r. This level set is just (\phi\circ \Phi^{-1}\circ\psi^{-1})(0)=\varphi(\Phi^{-1}(c)). By Theorem 14, the level set S is an embedded submanifold of dimension m-r, where m=\dim M.

Let c be a regular value. Proposition 3 implies that the set of points where the differential is of full rank \dim N is open in M. Denote this set by U. Clearly \Phi^{-1}(c)\in U and \Phi| _ U is a submersion. By submersion level set theorem, the level set \Phi^{-1}(c) is an embedded submanifold of U. The conclusion \Phi^{-1}(c) is an embedded submanifold of M is due to the fact that the composition of embeddings \Phi^{-1}(c)\hookrightarrow U\hookrightarrow M is still an embedding.
2.3. The tangent space to a submanifold Let S\subseteq M be an embedded submanifold. Then the inclusion map \iota:S\hookrightarrow M is an immersion, so we always have an injection \mathrm d\iota _ p:T _ pS\to T _ pM at every point p\in S. For v\in T _ pS and \tilde v=\mathrm d\iota _ p(v)\in T _ pM, \mathrm d\iota _ p acts in the following way: \tilde v(f)=v(f\circ\iota)=v(f| _ S).

We can identify T _ pS with its image under the linear map \mathrm d\iota _ p, so that it is a linear subspace of T _ pM.

The next proposition gives us a characterization of T _ pS.

Proof. Let v\in T _ pM such that there is a w\in T _ pS satisfying v=\mathrm d\iota _ p(w). If f is a smooth function vanishing on S, then vf=v(f| _ S)=0. Conversely if v\in T _ p M satisfies vf=0 for any smooth function f vanishing on S, we need to find the corresponding tangent vecor in T _ pS. There is a chart (U,\varphi) containing p such that all but the first k coordinates vanish on U\cap S. Since T _ pM is spanned by \frac{\partial}{\partial x^1}| _ p,\dots,\frac{\partial}{\partial x^n}| _ p, we can know T _ p S is spanned by the first k of \frac{\partial}{\partial x^i}| _ p. Therefore, v=v^i\frac{\partial}{\partial x^i}| _ p\in T _ pS iff v^{k+1}=\dots=v^n=0.

Let \psi be a smooth bump function supported in U that is equal to 1 in a neighborhood of p and consider the smooth function f(x)=\psi(x)x^{k+1} which satisfies f| _ S=0. Then
\[0=vf=v^i\frac{\partial}{\partial x^i}\Big| _ p(\psi(x)x^{k+1})=v^i\frac{\partial(\psi(x)x^{k+1})}{\partial x^i}(p)=v^{k+1}.\]Similarly v^{k+2}=\dots=v^n=0. Thus v\in T _ pS.

Whitney Embedding Theorem

Finally in this chapter, let's take a look at the Whitney embedding theorem, saying every smooth manifold can be embedded in some Euclidean space. This justifies our habit of visualizing manifolds as subsets of \mathbb R^n.