====== 2020-01-29, Wednesday ======
$$\gdef\d\partial, \gdef\t\tilde$$

We first introduce the notion of a tangent bundle. Next, we mention that a smooth map $f: M \to N$ induces a global differential $df: TM \to TN$. This finishes up Ch 3. Next, we introduce some more terminologies in Ch 4, Immersion and Submersion.

==== Tangent Bundle ====
Last time, we defined the 'setwise' tangent bundle as 
$$ TM = \bigsqcup_{p \in M} T_p M. $$
Elements of $TM$ is a pair $(p, v)$ where $p \in M$ and $v \in T_p M$. 
This $TM$ has a natural projection map 
$$\pi : TM \to M, \quad (p,v) \mapsto p$$. 

Now, we equip $TM$ with a smooth manifold structure by equip it with a coordinate charts. If $(U, \varphi)$ is a chart for $M$, we will define a chart $(\t U, \t \varphi)$ on $TM$, where $\t U := \pi^{-1}(U)$, and
$$  \t \varphi: \t U \mapsto \varphi(U) \times \R^n \subset \R^n \times \R^n. $$
Let $u^i = x^i \circ \varphi: U \to \R$ be components of $\varphi$, recall that $\{ \frac{\d}{d u^i}(p) \}_{i=1}^n$ is a basis for $T_p M$, thus we may decompose $v \in T_p M$ as 
$$ v = \sum_{i=1}^n v^i \frac{\d}{d u^i}(p). $$
Thus, we can define $\t \varphi$ as 
$$\t \varphi: (p, v) \mapsto (u^1, \cdots, u^n; v^1, \cdots, v^n). $$

Suppose we have two such charts $(U_1, \varphi_1)$ and $(U_2, \varphi_2)$, we want to check that on $\t U_1 \cap \t U_2$, the transition map 
$$ \t \varphi_2 \circ \t \varphi_1^{-1}: \t \varphi_1( \t U_1 \cap \t U_2) \to \t \varphi_2( \t U_1 \cap \t U_2)$$
is a diffeomorphism. The base coordinates $u^i$ changes according to $\varphi_2 \circ \varphi_1^{-1}$, which is a diffeomorphism, the fiber coordinates $v^i$ changes as 
$$ v^i_2  = \sum_{j=1}^n \frac{\d u_2^i }{\d u_1^j } v^j_1 \quad \Leftarrow \quad v = \sum_i v_1^i \frac{\d }{\d u_1^i} = \sum_i v_2^i \frac{\d }{\d u_2^i}$$
Since the Jacobian matrices $\frac{\d u_2^i }{\d u_1^j } $ are smooth and invertible, we see $\t \varphi_2 \circ \t \varphi_1^{-1}$ is smooth. It is not hard to verify its inverse is smooth as well. Thus, we have produced from each coordinate chart $(U, \varphi)$ on $M$  a coordinate chart $(\t U, \t \varphi)$ on $TM$, thus giving $TM$ a smooth manifold structure. 
See Lee p63 for details. 

==== Global differential ====
Last time, we defined pointwise linear map $df(p): T_p M \to T_{f(p)} N$. Put them together, we get a global differential $df: TM \to TN$. To verify that this map is a smooth map, we just need to work in local coordinate chart, and verify that the corresponding map between charts is smooth. 

===== Immersion and submersions =====
==== Implicit Function Theorem, Inverse Function Theorem ====
We recall the following results from calculus. 

**Inverse Function Theorem**. If $f: \R^n \to \R^n$ is smooth, $f(0)=0$, and $df(0)$ is invertible, then there exists a neighborhood $U$ of $0$, such that $f|_U$ is invertible with smooth inverse. 


**Implicit Function Theorem**. Let $f: \R^n_x \times \R_y^m \to \R_z^m$ be a smooth function, where subscripts $x,y,z$ are coordinates on the corresponding factors.  Assume that $f(0,0) = 0$,  and the (partial) Jacobian matrix $\frac{\d f}{\d y} \vert_{(0,0)}$ is invertible. Then, there exists a neighborhood $0 \in U \subset \R^n_x$, and a smooth map $g:U \to \R^m_y$, such that $g(0)=0$ and $f(x,g(x))=0$ for all $x \in U$. 


==== Local property of $f$ is determined by $df$ ====
The rank of a map at point $p$ is the rank of the linear map $df(p): T_p M \to T_{f(p)} N$. More concretely, if we put coordinate charts around $p$ and $f(p)$, then $df(p)$ is represented as the Jacobian matrix. 

We only consider constant rank map, that is $r=rank(df(p))$ is constant on $M$. There are several situtations
  * If $r = \dim M$, i.e. $df$ is injective, we call $f$ an //immersion//. 
  * If $r = \dim N$, i.e. $df$ is surjective, we call $f$ an //submersion//.
  * If $r = \dim M = \dim N$, $df$ is bijective, we call $f$ a //local diffeomorphism//. 
Note that these are local properties of $f$. 

In general, we have the rank theorem (Lee Thm 4.12, p81), which says $f$ can factorize as a submersion followed by an immersion 
$$\gdef\into\hookrightarrow \gdef\onto\twoheadrightarrow$$
$$f: M \onto Z \into N$$