5. Submanifold

5.1Submanifold

Recall $M$ is a smooth manifold and $U\subset M$ open $\Rightarrow$ $U$ is a smooth manifold.

We want a broader definition of submanifold, e.g. incorporating things like $S^n\subset {\mathbb{R}}^{n+1}$ or

定义 5.1.1. A subset $N\subset M$, where $M$ is a smooth manifold called a submanifold if for every point $p\in N$, there exists a chart for $M$, centered at $p$, say $(U,\phi )$, such that$$\phi (N\cap U)=\{x_1=\cdots =x_k=0\}\cap \phi (U)\subset {\mathbb{R}}^m$$where $\dim M=m$ and $k$ is some fixed non-negative integer.

注 5.1.2. Clearly, $\dim N=m-k⩽m=\dim M$.

5.2Immersion, Submersion and Embedding

定义 5.2.1. Let $f:M\to N$ be a map of smooth manifolds and $p\in M$.

(1)	$f$ is called an immersion at $p$, if $D_{p}f:T_{p}M\to T_{f(p)}N$ is injective.
(2)	$f$ is called a submersion at $p$, if $D_{p}f$ is surjective.
(3)	$f$ is called an immersion/submersion, if it is an immersion/submersion at all points in $M$.
(4)	$f$ is called an embedding, if it is an immersion and a homeomorphism onto its image.

例 5.2.2.  

(1)	$\begin{array}{rl}i:{\mathbb{R}}^m& \to {\mathbb{R}}^n\text{with n⩾m is an immersion (and an embedding).}\\ (x_1,\dots ,x_m)& \mapsto (x_1,\dots ,x_m,0,\dots ,0)\end{array}$
(2)	$\begin{array}{rl}\pi :{\mathbb{R}}^m& \to {\mathbb{R}}^n\text{with}m⩾n\text{is a submersion.}\\ (x_1,\dots ,x_n,x_{n+1},\dots ,x_m)& \mapsto (x_1,\dots ,x_n)\end{array}$
(3)	$(a,b)\stackrel{\gamma }{\to }{\mathbb{R}}^2$ is an immersion but not an embedding. Similarly, is an immersion but not an embedding.

注 5.2.3. If $f:M\to N$ is an immersion, $\dim M⩽\dim N$. If $f:M\to N$ is a submersion, $\dim M⩾\dim N$.

定理 5.2.4. Let $f:M\to N$ be an immersion at $p\in M$. Then, there exist charts $(U,\phi )$ around $p$ and $(V,\psi )$ around $q=f(p)$, s.t. $\psi \circ f\circ {\phi }^{-1}=i{|}_{\phi (U)}$, i.e. $\psi \circ f\circ {\phi }^{-1}(x_1,\dots ,x_m)=(x_1,\dots ,x_m,0,\dots ,0)$.

证明. Take charts $(U_0,{\phi }_0)$ around $p$ and $(V_0,{\psi }_0)$ around $q$. The Jacobi matrix of ${\psi }_0\circ f\circ {\phi }_0^{-1}$ at $0$ has rank $m=\dim M$ by assumption. After reordering the coordinates of ${\psi }_0$, we obtain a new chart $(V_0,\psi )$, s.t. for $F=\psi \circ f\circ {\phi }^{-1}$, ${\left(\frac{\partial F_i}{\partial x_j}\right)}_{\text{ij=1,…,m=1,…,m}}$ is invertible.

$\begin{array}{rl}\text{Now define}G:\phi (U_0)\times {\mathbb{R}}^{n\times m}& \to {\mathbb{R}}^n\\ (x_1,\dots ,x_m,x_{m+1},\dots ,x_n)& \mapsto F(x_1,\dots ,x_m)+(0,\dots ,0,x_{m+1},\dots ,x_n)\end{array}$

$D_{0}G=\left(\begin{array}{cc}{\left(\frac{\partial F_i}{\partial x_j}\right)}_{i,j=1}^m& \ast \\ 0& \mathrm{Id}\end{array}\right)$ is invertible. By the inverse function theorem, we find$$0\in \phi (U)\underset{\text{open}}{\subseteq }\phi (U_0)0\in \psi (V)\underset{\text{open}}{\subset }\psi (V_0)$$and a smooth function $H$$$H:\psi (V)\to \phi (U)\times U_1$$s.t. $G\circ H=\mathrm{Id}$ and $H\circ G=\mathrm{Id}$ where defined.

Now set $\tilde{\psi }=H\circ \psi$, then $\tilde{\psi }\circ f\circ {\phi }_0^{-1}=H\circ F=H\circ G\circ i=i$.

$\square$

注 5.2.5. We only needed to modify the chart for the target.

We also have

定理 5.2.6. If $f:M\to N$ is a submersion ($m⩾n$) at $p\in M$, there are charts $(U,\phi )$ around $p$ and $(V,\psi )$ around $q=f(p)$, s.t. $\psi \circ f\circ {\phi }^{-1}(x_1,\dots ,x_m)=(x_1,\dots ,x_n)$.

证明. Take arbitrary charts $(V,\psi )$ and $(U_0,{\phi }_0)$ around $q$, respectively $p$. After reordering coordinates of ${\phi }_0$, we may assume for $F=\psi \circ f\circ {\phi }_0^{-1}$, we have ${\left(\frac{\partial F_i}{\partial x_j}\right)}_{i,j=1}^n$ is invertible. Define$$G(x_1,\dots ,x_n,x_{n+1},\dots ,x_m)=(F_1(x_1,\dots ,x_m),\dots ,F_n(x_1,\dots ,x_m),x_{n+1},\dots ,x_m)$$Then$$D_{0}G=\left(\begin{array}{cc}{\left(\frac{\partial F_i}{\partial x_j}\right)}_{i,j=1}^n& \ast \\ 0& \mathrm{Id}\end{array}\right)$$is invertible, so we have a local inverse $H$ (possibly shrinking the domain of definition). Then set $\phi =G\circ {\phi }_0$ where defined. This gives$$\psi \circ f\circ {\phi }^{-1}=\psi \circ f\circ {\phi }_0^{-1}\circ G^{-1}=F\circ H=\pi \circ G\circ H=\pi$$

$\square$

定理 5.2.7. Let $f:M\to N$ be an embedding. Then $f(M)\subset N$ is a submanifold.

证明. Let $q\in f(M)$. Because $f$ is a homeomorphism onto its image, there is a unique preimage $p$, s.t. $f(p)=q$ and a chart centered at $q$, say $(V,\psi )$, s.t. $f^{-1}(V)=U\subset M$ admits a chart $\phi$. Arguing as in the previous theorem, we can assume $(\psi \circ f\circ {\phi }^{-1})(x_1,\dots ,x_n)=(x_1,\dots ,x_m,0,\dots ,0)$, thus $\psi (f(M)\cap V)=\{x_{m+1}=\cdots =x_n=0\}\cap \phi (V)$.

$\square$

注 5.2.8. Conversely, for any submanifold $Z\subset N$, the inclusion $Z\subset N$ is an embedding.

5.3Regular Value

定义 5.3.1. Let $f:M\to N$ be a map of manifolds, $q\in N$ is called regular value if all points $p\in f^{-1}(q)$ satisfy that $D_{p}f$ are surjective.

注 5.3.2. By a theorem of Sard, the set of regular values of a map is dense (in $N$).

Fact (Sard’s Theorem). The set of regular values of a smooth map is dense in the target manifold.

例 5.3.3. If $\dim M<\dim N$,

•	every point not in the image of $f$ is a regular value (this always holds);
•	every point in the image of $f$ is not a regular value.

例 5.3.4. Let $\begin{array}{rl}f:{\mathbb{R}}^2& \to \mathbb{R}\\ (x,y)& \mapsto x\cdot y\end{array}$ $⇝$ $\begin{array}{rl}{D}_{(x,y)}& f:{\mathbb{R}}^2\to \mathbb{R}\\ =& (b,a)\text{has full rank iff}(a,b)\ne (0,0)\end{array}$

定理 5.3.5. If $f:M\to N$ is smooth and $p\in N$ is regular value, then $f^{-1}(p)$ is a submanifold of $M$.

证明. Let $q\in f^{-1}(p)$. Then by the local form for submersions, we find charts $(U,\phi )$, $(V,\psi )$ around $q$, $p$, s.t. $\psi \circ f\circ {\phi }^{-1}(x_1,\dots ,x_m)=(x_1,\dots ,x_n)$ is the projection. But then $\phi (f^{-1}(p)\cap U)=\{x_1=\cdots =x_n=0\}\cap \phi (U)$.

$\square$

5.4Whitney’s Embedding Theorem

定理 5.4.1 (Whitney’s Embedding Theorem). Every smooth manifold of dimension $n$ can be embedded into ${\mathbb{R}}^{2n}$.

注 5.4.2.  

•	In general, this dimension is optimal, e.g. non-orientable surfaces (${\mathbb{RP}}^2$, Klein bottle) cannot be embedded into ${\mathbb{R}}^3$ (but immersed). For particular manifold, better bonds on the dimension are possible, e.g. $S^1\times S^1\hookrightarrow {\mathbb{R}}^3$ or ${\mathbb{R}}^2\stackrel{\mathrm{Id}}{\hookrightarrow }{\mathbb{R}}^2$.
•	Any $m$-dimensional manifold can be immersed into ${\mathbb{R}}^{2m-a(m)}$, where $a(2m)$ is the number at $1$’s in the binary expansion of $m$.

We will only prove the following weaker version.

定理 5.4.3 (Weak Whitney’s Theorem). Every compact $m$-dimensional smooth manifold can be embedded into ${\mathbb{R}}^{2m+1}$.

证明. Let $X$ be a compact smooth $m$-dimensional manifold.

Claim 5.4.4. $X$ can be embedded into some ${\mathbb{R}}^k$ for $k\gg 0$.

证明. Let $\{(U_i,{\phi }_i){\}}_{i=1}^n$ be a finite atlas for $X$. Choose a partition of unity $\{{\rho }_i\}$ subordinate to $\{U_i{\}}_{i=1}^n$.

Next, define $\begin{array}{rl}\varphi :X& \to {\mathbb{R}}^k\text{with}k=n(m+1)\\ p& \mapsto ({\rho }_1(p)\cdot {\phi }_1(p),\dots ,{\rho }_n(p)\cdot {\phi }_n(p),{\rho }_1(p),\dots ,{\rho }_n(p))\end{array}$
Then $\varphi$ is an embedding.

$\square$

In fact, $\mathbf{\varphi }$ is injective: Let $\varphi (p_1)=\varphi (p_2)$. Choose $i$, s.t. $${\rho }_i(p_1)={\rho }_i(p_2)\ne 0$$Then$$\begin{array}{rl}{\rho }_i(p_1)\cdot {\phi }_i(p_1)={\rho }_i(p_2)\cdot {\phi }_i(p_2)& \Rightarrow {\phi }_i(p_1)={\phi }_i(p_2)\\ & \stackrel{{\phi }_i\text{different}}{\Rightarrow }{p}_1=p_2\end{array}$$${\mathbf{D}}_{\mathbf{p}}\mathbf{\varphi }$ is injective at all $\mathbf{p}\in \mathbf{M}$:$$\begin{array}{rl}& D_p\varphi :T_{p}X\to T_{\varphi (p)}{\mathbb{R}}^k\cong {\mathbb{R}}^k\\ & D_p\varphi =(D_p{\rho }_1\cdot {\phi }_1(p)+{\rho }_1(p)\cdot D_p{\phi }_1,\dots ,D_p{\rho }_n(p)+{\rho }_n(p)\cdot D_p{\phi }_n,D_p{\rho }_1,\dots ,D_p{\rho }_n)\end{array}$$Thus if $(D_p\varphi )(X)=0$ where $X\in T_{p}X$ $\begin{array}{rl}[t]& \Rightarrow (D_p{\rho }_i)(X)=0,\forall i\\ & \Rightarrow {\rho }_i(p)D_p{\phi }_i(X)=0,\forall i\\ & \stackrel{{\phi }_i\text{different}}{\Rightarrow }X=0\end{array}$
So $D_p{f}_i$ is injective.

引理 5.4.5. If $f:A\to B$ is an injective immersion of smooth manifold and $A$ is compact, then $f$ is an embedding.

证明. We need to show $f$ is a closed map.
If $Z\subset A$ is closed $\begin{array}{rl}& \stackrel{A\text{compact}}{\Rightarrow }Z\text{is compact}\\ & \stackrel{f\text{continuous}}{\Rightarrow }f(Z)\text{compact}\\ & \stackrel{B\text{Hausdorff}}{\Rightarrow }f(Z)\text{closed}\end{array}$

$\square$

Claim 5.4.6. If an $m$-manifold admits an injective immersion into ${\mathbb{R}}^k$ with $k>2m+1$, then it admits an injective immersion into ${\mathbb{R}}^{k-1}$.

证明. The idea is to project onto a generic hyperplane.

Hyperplanes are described via their normal vectors: For $[v]\in {\mathbb{RP}}^{k-1}$ denote by $P_{[v]}=\{u\in {\mathbb{R}}^k\mid ⟨u,v⟩=0\}$ the hyperplane orthogonal to $[v]$ and by ${\pi }_{[v]}:{\mathbb{R}}^k\to P_{[v]}$ the orthogonal projection.

Write ${\varphi }_{[v]}:={\pi }_{[v]}\circ \varphi :X\to {\mathbb{R}}^{k-1}$.

Claim: For a generic choice of $[v]$, ${\varphi }_{[v]}$ will be an injective immersion.

Assume ${\varphi }_{[v]}$ is not injective, i.e. there are $p_1\ne p_2\in X$, s.t. ${\varphi }_{[v]}(p_1)={\varphi }_{[v]}(p_2)$ and so $\varphi (p_1)-\varphi (p_2)$ lies in the line $[v]$, i.e. the points where ${\varphi }_{[v]}$ is not injective live in the image of$$\begin{array}{rl}(X\times X)\setminus {\Delta }_x& \to {\mathbb{RP}}^{k-1}\\ (p_1,p_2)& \mapsto [\varphi (p_1)-\varphi (p_2)]\end{array}$$where ${\Delta }_x=\{(x,x)\}$.

By Sard’s theorem, for a set containing an open dense set of $[v]$’s, ${\varphi }_{[v]}$ will be injective.

Similarly, consider a $[V]$, s.t. there exists $p\in X$ with $D_p{\varphi }_{[v]}$ not injective, i.e. there exists $0\ne A\in T_{p}X$, s.t. $\underset{D_p({\pi }_{[v]}\circ \varphi )}{\underbrace{D_p{\varphi }_{[v]}}}(A)=0$ $\iff$ $({\pi }_{[v]}\circ D_p\varphi )(A)$ $\iff$ $(D_p\varphi )(A)$ is contained in the line $[V]$.

注 5.4.7. $X\subset TX$ submanifold via $x\mapsto (x,0)$.

i.e. the $[v]$’s, s.t. ${\varphi }_{[v]}$ is not an immersion live in the image of$$\begin{array}{rl}TX\setminus X& \to {\mathbb{RP}}^{k-1}\\ (p,A)& \mapsto (D_p\varphi )(A)\end{array}$$where $p\in X$, $A\in T_{p}X$. Again by Sard’s theorem, the set s.t. ${\varphi }_{[v]}$ is an immersion, is open dense.

$\square$

Now take a $[V]$ in the intersection of these dense sets.

$\square$