2. Differentiable Manifold

2.1Charts

Locally Euclidean: $\forall x\in X$, $\exists U$ open and a homeomorphism $\phi :U\to V\subset {\mathbb{R}}^n$.

Define $(U_1,{\phi }_1)$ and $(U_2,{\phi }_2)$ as above. Then$${\phi }_2\circ {\phi }_1^{-1}:{\phi }_1(U_1\cap U_2)\stackrel{\text{homeomorphism}}{\to }{\phi }_2(U_1\cap U_2).$$The $(U_i,{\phi }_i)$ are called charts and $f_{21}={\phi }_2\circ {\phi }_1^{-1}$ is the transition map from the chart $(U_1,{\phi }_1)$ to the chart $(U_2,{\phi }_2)$.

2.2Atlas

定义 2.2.1. A collection of charts $(U_i,{\phi }_i)$, $i\in I$ with $\bigcup _{i\in I}{U}_i=M$ is called an atlas. We have the cocycle conditions/properties$$\begin{array}{rl}(1)& f_{ii}=\mathrm{Id}\\ (2)& f_{ij}=f_{ji}^{-1}\\ (3)& f_{ij}{f}_{jk}=f_{ik}\end{array}\forall i,j,k\in I$$The $f_{ij}$ for pairs $i,j\in I$ with $U_i\cap U_j\ne \varnothing$ form the structure cocycle of the given atlas$$\mathcal{A}=\{(U_i,{\phi }_i)\mid i\in I\}.$$

命题 2.2.2. Let $\mathcal{A}$ be an atlas for $M$. From the collection of open subsets $V_i={\phi }_i(U_i)\subset {\mathbb{R}}^n$ together with the structure cocycle, one can reconstruct $M$.

证明. $\overline{M}=\left(\coprod _{i\in I}{V}_i\right)/\sim$, where $\sim$ is the equivalence relation given by $V_i\ni p\sim q=f_{ji}(p)\in V_j$, $\forall i,j\in I$.

$$\begin{array}{rl}a:\overline{M}& \to M\\ [p]& \mapsto {\phi }_i^{-1}(p)\text{if}p\in V_i\end{array}$$If $q\in V_j$ is equivalent to $p$, then $q=f_{ji}(p)$ $\Rightarrow$ ${\phi }_j^{-1}(q)={\phi }_j^{-1}({\phi }_j^{-1}{\phi }_i^{-1})(p)={\phi }_i^{-1}(p)$. So $a$ is well-defined. $a$ is also continuous.$$\begin{array}{rl}b:M& \to \overline{M}\\ m& \mapsto [{\phi }_i(m)]\text{if}m\in U_i\end{array}$$If $m$ is also in $U_j$, then ${\phi }_j(m)=({\phi }_j\circ {\phi }_i^{-1}){\phi }_i(m)=f_{ji}({\phi }_i(m))$. So $b$ is well-defined. $b{|}_{U_i}=\pi \circ {\phi }_i$, where $\pi :\coprod _{i\in I}{V}_i\to M$ is the projection onto equivalent classes. Thus $b$ is continuous.

$$\begin{array}{llcllll}\overline{M}& \stackrel{a}{\to }& M& \stackrel{b}{\to }& \overline{M}& & \\ [p]& \mapsto & {\phi }_i^{-1}(p)& \mapsto & [{\phi }_i{\phi }_i^{-1}(p)]=[p]& \Rightarrow & b\circ a={\mathrm{Id}}_{\overline{M}}\\ M& \stackrel{b}{\to }& \overline{M}& \stackrel{a}{\to }& M& & \\ m& \mapsto & [{\phi }_i(m)]& \mapsto & {\phi }_i^{-1}{\phi }_i(m)=m& \Rightarrow & a\circ b={\mathrm{Id}}_M\end{array}$$

$\square$

2.3Differentiable Manifold

定义 2.3.1. A smooth or differentiable manifold is a topological manifold together with an atlas $\mathcal{A}$ for which $f_{ij}$ are smooth/differentiable.

Smooth means ${\mathcal{C}}^r$ for some $r⩾2$.

Terminology. Such an atlas is called a smooth atlas. Two smooth atlases $\begin{array}{rl}{\mathcal{A}}_1& =\{(U_i,{\phi }_i)\mid i\in I\}\\ {\mathcal{A}}_2& =\{(U_k^{\prime },{\phi }_k^{\prime })\mid k\in I^{\prime }\}\end{array}$ on $M$ are equivalent if ${\mathcal{A}}_1\cup {\mathcal{A}}_2$ is also a smooth atlas.

2.4Differentiable Structure

定义 2.4.1. A differentiable structure on $M$ is a maximal smooth atlas, equivalently an equivalence class of atlases for the above.

Fact. Every maximal ${\mathcal{C}}^r$ atlas contains a unique maximal ${\mathcal{C}}^{\infty }$ atlas. Because of this, we will only consider ${\mathcal{C}}^{\infty }$ manifolds.$$\text{smooth}=\text{differentiable}={\mathcal{C}}^{\infty }$$

定义 2.4.2. Let $M$ and $N$ be smooth manifolds, $f:M\to N$ is smooth if $\forall p\in M$, $\exists$ a chart $(U,\phi )$ with $p\in U$ and a chart $(V,\psi )$ for $N$ with $f(p)\in V$ such that $\psi \circ f\circ {\phi }^{-1}$ is smooth.

例 2.4.3. $f:M\to \mathbb{R}$ is smooth if and only if $f\circ {\phi }^{-1}$ is smooth for all charts $(U,\phi )$.

定义 2.4.4. $f:M\to N$ is a diffeomorphism if it is bijective, differentiable, and $f^{-1}$ is also differentiable.

例 2.4.5. Every $B(x,\epsilon )\subset {\mathbb{R}}^n$ is diffeomorphic to ${\mathbb{R}}^n$.

注 2.4.6. Not every topological manifold has a differentiable structure. If it has one, it may fail to be unique!

For $n⩽3$, every topological manifold has a differentiable structure, unique up to diffeomorphism.

For $n⩾4$, there are manifolds with no differentiable structure, and there are manifold with unusual non-diffeomorphic differentiable structures.

例 2.4.7. The topological manifold ${\mathbb{R}}^4$ has infinitely many distinct differentiable structures.

例 2.4.8. $S^7$ has several distinct differentiable structures.

2.5“The Smooth Invariance of Domain”

Differentiable atlas means that transition functions between charts are diffeomorphisms. The way we had defined differentiable manifolds, we assume that we always have a fixed dimension, so we define a manifold of dimension $n$ which is locally homeomorphic to ${\mathbb{R}}^n$. Now we want to show that in the differentiable case, functions as dimension given are actually redundant.

Take a manifold $M$. Assume we have two charts $U_1$, $U_2$, and ${\phi }_1:U_1\to V_1\subset {\mathbb{R}}^m$ and ${\phi }_2:U_2\to V_2\subset {\mathbb{R}}^n$.

Then we have a transition map $f_{21}={\phi }_2\circ {\phi }_1^{-1}:{\phi }_1(U_1\cap U_2)\to {\phi }_2(U_1\cap U_2)$.

If the transition map $f_{12}$, $f_{21}$ are diffeomorphisms, then $m=n$. Since$$\begin{array}{r}{f}_{12}\circ f_{21}={\mathrm{Id}}_{{\phi }_1(U_1\cap U_2)}\\ f_{21}\circ f_{12}={\mathrm{Id}}_{{\phi }_2(U_1\cap U_2)}\end{array}\stackrel{differentiate}{⇝}\begin{array}{r}{D}_{{\phi }_2(x)}{f}_{12}\circ D_{{\phi }_1(x)}{f}_{21}={\mathrm{Id}}_{{\mathbb{R}}^m}\\ D_{{\phi }_1(x)}{f}_{21}\circ D_{{\phi }_2(x)}{f}_{12}={\mathrm{Id}}_{{\mathbb{R}}^n}\end{array}$$Both derivatives on the LHS are isomorphisms.which implies$$m=n$$This is called “the smooth invariance of domain”.

Given a smooth manifold $M$ with a smooth atlas $(U_i,{\phi }_i)$, $i\in I$, we can reconstruct $M$ up to diffeomorphism just from ${\phi }_i(U_i)$, $i\in I$, together with the structure cocycle given by the transition function $f_{ij}$.