1. Tangent spaces and the rank theorem.
Definition 1.1. Let $M$ be an $m$-dimensional differentiable manifolds, and $\gamma_i: (-\epsilon,\epsilon)\to M$, $i=1,2$ be two differentiable paths on $M$, such that $\gamma_1(0)=\gamma_2(0)=a\in M$. We define an equivalent relation between $\gamma_1,\gamma_2$, denoted by $\gamma_1\sim\gamma_2$ if there exists a chart $(U,\phi)$ around $a$ such that
$$\frac{d}{dt}\phi\circ\gamma_1(t)|_{t=0}=\frac{d}{dt}\phi\circ\gamma_2(t)|_{t=0}$$
All equivalent classes of such paths form a tangent space of $M$ at $a$. We denote this as $T_aM$.
Another realization of tangent space is to look at all germs of differtiable functions at $a$. We denote this sets as $\mathscr{E}_{M,a}$. If we choose a chart $(U,\phi)$ around $a$ such that $\phi(a)=(0,...,0)\in \mathbb{R}^m$, then it can be seen that $\mathscr{E}_{M,a}$ can be identified with $\mathscr{E}_{\mathbb{R}^m,0}$.
A derivative is a $\mathbb{R}$-linear map $\delta$ from $\mathscr{E}_{M,a}$ to $\mathbb{R}$, such that:
$$\sigma(fg)=\sigma(f)g(a)+\sigma(g)f(a)$$.
All derivatives from $\mathscr{E}_{M,a}$ to $\mathbb{R}$ form a $\mathbb{R}$-vector space, and we denote this as $Der_{M,a}$. It can be seen that $Der_{M,a}$ is actually isomorphic to $Der_{\mathbb{R}^m,0}$. And the latter one is an $m$-dimensional $\mathbb{R}$-vector space, with basis $(\frac{\partial }{\partial x_1},...,\frac{\partial}{\partial x_m})$. Hence, we can see that in fact $Der_{M,a}$ is an $m$-dimensional $\mathbb{R}$-vector space.
We next want to identify $T_aM$ with $Der_{M,a}$. For any $\gamma:(-\epsilon,\epsilon)\to M$ is a differentiable path, with $\gamma(0)=a$. We can construct a derivative $\delta_\gamma$ as follows.
$\delta_\gamma(f)=\frac{d}{dt}f\circ\gamma(t)|_{t=0}$
for any differentiable map $f$ in $\mathscr{E}_{M,a}$. We can see that it is a well-defined map from $T_aM$ to $Der_{M,a}$, and is bijective. Hence, $T_aM$ can be considered as an $m$-dimensional $\mathbb{R}$-vector space, where $m$ is the dimension of $M$.
The following notions is important.
If $f: M\to N$ be a differentiable map between two differentiable manifolds $M, N$. Then one can see $f$ induces a ring homomorphism $f^*: \mathscr{E}_{N,f(a)}\to \mathscr{E}_{M,a}$, for any $a\in M$. From this, one can define the map $T_af: T_aM\to T_{f(a)}N$ by sending $\delta$ to $\delta\circ f^*$. It is an $\mathbb{R}$-linear map. We define $rank_a(f):=rank(T_af)$. The rank theorem is stated as follows.
Theorem 1.2 (Rank Theorem). Let $f: M\to N$ be differentiable map between differentiable manifolds, with $a\in M$, and $r=rank_a(f)$. Then there exists a chart $(U,\phi)$ around $a$, and a chart $(V,\psi)$ around $f(a)$ such that the map $\psi\circ f\circ \phi^{-1}$ from $\mathbb{R}^m$ to $\mathbb{R}^n$ is given by sending $(x_1,...x_r,...,x_m)$ to $(x_1,...,x_r,0...,0)$.
2. Transversality and applications of rank theorem.
If $f: M\to N$ is defined as above, and there exists a such that $T_af:T_aM\to T_aN$ is injective. then because $\dim_\mathbb{R}(T_aM)=\dim M$, and $\dim_{\mathbb{R}}(T_{f(a)}N)=\dim N$, we have $\dim M\le \dim N$ in this case. Similarly, if there exists $a\in M$ such that $T_af$ is surjective. then $\dim M\ge \dim N$. If for all $a\in M$, $T_af$ is injective, $f$ is called immersive. And if $T_af$ is surjective for all $a\in M$, $f$ is called submersive. For any $q\in N$, and $q\in f(M)$, if for all $p\in f^{-1}(q)$, $T_pf$ is surjective, we call $q$ is a regular value of $f$.
We also need the notion of submanifolds.
Definition 2.1. Let $M$ be an $m$-dimensional differentiable manifold, and $N$ is a closed subset of $N$. We call $N$ is an $n$-dimensional submanifold of $M$ if for any $p\in N$, there exists a chart $(U,\phi)$ of $M$ around $p$ such that $\phi(N\cap U)=\phi(N)\cap \mathbb{R}^n$, where $R^n$ is identified with a subspace of $\mathbb{R}^m$ with some coordinates are zero.
The following lemma is a surprising fact.
Lemma 2.2. If $f:M\to N$ be differentiable map between manifolds defined as above, with $\dim M=m,\dim N=n$. And $q\in f(M)$ is a critical value of $f$. then $f^{-1}(q)$ is an $m-n$ dimensional submanifold of $M$.
Proof. It is just a definition of the rank theorem. We can choose line bundle charts $(U,\phi),(V,\psi)$ around $q$, and $p\in f^{-1}(q)$, respectively, such that $\psi\circ f\circ\phi^{-1}(x_1,...,x_m)=(x_1,...,x_n)$, and that $\phi(p)=(0,...,0)\in\mathbb{R}^m. \psi(q)=(0,...,0)\in \mathbb{R}^n$. Then one can see that $\phi(f^{-1}(q)\cap U)=\phi(f^{-1}(q))\cap \mathbb{R}^{m-n}$ (note that $f^{-1}(q)$, as viewed in the real coordinates, is just the inverse image of $(0,...,0)\in \mathbb{R}^n$, which is $(0,...,0,x_{n+1},...,x_m)$, which can be identified with $\mathbb{R}^{m-n}$). (Q.E.D)
We now come to an important notion about transversality.
Definition 2.3. Let $f: M\to N$ be a differtiable map between differentiable manifolds, and $L$ is a submanifold of $N$, with $j: L\to N$ be the natural inclusion. For any $p\in f^{-1}(L)$, if $T_pf(T_pM)+T_{f(p)}j(T_{f(p)}L)=T_{f(p)}N$, then $f$ and $L$ is called transverse of each other.
We now explain what the definition means. First, we can consider $T_{f(p)}L$ as a subspace of $T_{f(p)}N$, and hence $T_pf$ maps surjectively to $T_{f(p)}N/T_{f(p)}N$. But then, locally around $f(p)$, by defintion of submanifold, we can consider $L$ as $X\subset \mathbb{R}^l$, and hence, $N$ as $X\times Y$ (via chart $(W,\psi)$), where $Y\subset \mathbb{R}^{n-l}$. From this, $T_{f(p)}N/T_{f(p)}L$ can be realized by $T_0Y$.
Also, locally at $p$, one can view $M$ as an open subset $U'$ of $\mathbb{R}^n$ around $0$ via chart $(U,\phi)$, with $\phi(p)=0\in \mathbb{R}^n$. And $U'$ is map to $Y$ via the map
$$g:U'\xrightarrow{\psi\circ f\circ \phi^{-1}} X\times Y\xrightarrow{\pi} Y$$
And hence, via $g$, $T_0U' (\equiv T_pM)$ is mapped surjectively to $T_0Y$. By the definition of regular value, we can see that $0$ is a regular value of $g$. By Lemma 2.2, $g^{-1}(0)=f^{-1}(L)$ is a submanifold of $M$, and
$$\dim f^{-1}(L)=\dim f^{-1}(X\times 0)=\dim g^{-1}(0)=\dim M-(\dim N-\dim L)$$
This is equivalent to $codim(f^{-1}(L))=codim(L)$. We obtain a following
Theorem 2.4. Let $f: M\to N$ be a differentiable map between differentiable manifolds, and $L\subset N$ be a submanifold such that $f$ and $L$ are transverse of each other. Then $codim(f^{-1}L)=codim(L)$.
3. Tangent bundles and vector fields. We first need the notion of vector bundles.
Definition 3.1. Let $E, M$ be two differentiable manifolds, together with a (projection) differentiable map $\pi; E\to M$. Then $(E,\pi)$ is called a vector bundle of rank $n$ over $M$ if the following conditions holds.
i. $\pi^{-1}(U_i)$ is homeomorphic to $U_i\times \mathbb{R}^n$, via a homeomophism $\pi_i$ where $\{U_i\}$ is an open covering of $M$.
ii. $\pi_i\circ pr_1=\pi$, where $pr_1$ is a projection map from $U_i\cap \mathbb{R}^n\to U_i$.
iii. For any $p\in M$, $\pi^{-1}(p)\cong \{p\}\times \mathbb{R}^n$ as $\mathbb{R}$-vector spaces.
We are now ready to construct the tangent bundle of $M$, where $M$ is an $m$-dimensional differentiable manifold. Let $TM=\cup_{a\in M}T_aM$. We consider the map $\pi: TM\to M$, by sending an element in $T_aM$ to $a$. One can see that for any $a\in M$. $f^{-1}(a)=T_aM$. which is isomorphic to $\{a\}\times \mathbb{R}^m$. Let $U$ be any open subset of $M$, one can see easily that there exists a bijective map between $\pi^{-1}(U)$ and $U\times\mathbb{R}^m$. And hence, for any chart $\phi_i: U_i\to U_i'$, we can equip a corresponding chart for $TM$, $\psi_i: \pi^{-1}(U_i)\to U_i'\times \mathbb{R}^m$, and transition functions between charts induced from transition functions between charts on $M$. The topological structure on $M$ is defined as follows. $U\subset TM$ is open if $\psi_i(U\cap \pi^{-1}(U_i))$ is open in $U_i'\times \mathbb{R}^m$.
By this construction. $TM$ now becomes a differentiable manifold, and $\pi: TM\to M$ is a differentiable map satisfying Definition 3.1. This implies $TM$ is a vector bundle of rank $m$ over $M$.
Definition 3.2. A section of $E$ over $M$ is a differentiable map $\sigma$ from $M$ to $E$, such that $s(p)$ is mapped to $E_p$, the fiber of $p$, i.e. $\pi^{-1}(p)$.
Note that the definition above is equivalent to that $\sigma: M\to E$ is a differentiable map with $\pi\circ\sigma =id_M$. Now, a vector field $X$ on $M$ is a section $X: M\to TM$.
Definition 3.2. A section of $E$ over $M$ is a differentiable map $\sigma$ from $M$ to $E$, such that $s(p)$ is mapped to $E_p$, the fiber of $p$, i.e. $\pi^{-1}(p)$.
Note that the definition above is equivalent to that $\sigma: M\to E$ is a differentiable map with $\pi\circ\sigma =id_M$. Now, a vector field $X$ on $M$ is a section $X: M\to TM$.
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