$\gdef\d\partial, \gdef\t\tilde$

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.

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.

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.

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 $x \in U$.

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 $\gdef\into\hookrightarrow \gdef\onto\twoheadrightarrow$ $f: M \onto Z \into N$