Lecture Notes

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====== 2020-02-14, Friday ======

===== Vector Bundle =====
** Definition ** : A vector bundle is a quadruple $(E, \pi, M ,F)$, such that 
  * $E, M$ are smooth manifolds
  * $\pi: E \to M$ is a surjective submersion. For each $U \subset M$, we set $E|_U := \pi^{-1}(U)$. 
  * $F$ is a vector space of rank $n$ over $\R$. 
  * there exists a trivializing cover, ie. an open cover $\gdef\cU{\mathcal U} \cU$ of $M$, and for every $U \in \cU$ a diffeomorphism $$ \Psi_U: E|_U \to U \times F, \quad v \mapsto (p=\pi(v), \Phi^U_p(v)) $$
    * $\Phi^U_p: E_p \to F$ is a diffeomorphism for any $p \in U$. 
    * If $U, V \in \cU$ are two trivializing neighborhoods with non-empty overlap $U \cap V$, then, the map $\Phi_{VU}(p): \Phi^V_p \circ (\Phi^U_p)^{-1}: F \to F$ is a linear isomorphism. And moreover, the map $$p \mapsto \Phi_{VU}(p) \in GL(n, \R)$$ is smooth (i.e. each entry of hte matrix is a smooth function of $p$)

Given a smooth manifold $M$, and fiber $F = \R^n$, how to specify the data of a vector bundle? We just need to specify a cover $\cU$ and some gluing data: $$g_{\alpha\beta} = g_{\alpha \gets \beta}: U_\alpha \cap U_\beta \to Aut(F)=GL(\R,n)$$ for any $U_\alpha ,U_\beta \in \cU$, that satisfies the tricycle condition
  - $g_{\alpha \beta} \circ g_{\beta \alpha}=1_F$ over $U_\alpha \cap U_\beta$
  - $g_{\alpha \beta} \circ g_{\beta\gamma} \circ g_{\gamma \alpha} = 1_F$ over $U_\alpha \cap U_\beta \cap U_\gamma$. 

===== Cotangent Bundle =====
Some linear algebra first. Let $V$ be a finite dimensional vector space over $\R$

** Dual vector space of $V$ ** is the vector space of linear function on $V$, denoted as $V^*$. 

If $V$ is a basis $\{E_1, \cdots, E_n\}$, then there is a basis basis $\{ \epsilon^i\} $ of $V^*$, called **dual basis to $\{E_i\}$**, satisfying $\epsilon^i( E_j) = \delta_j^i$. 

Given a basis for $V$ and corresponding dual basis as above, then a vector $v \in V$ and a covector $\omega \in V^*$ can be written as 
$$ v = \sum_i v^i E_i , \quad \omega = \sum_j \omega_j \epsilon^j $$
The canonical pairing $(\omega, v) = \omega(v) $ can be written as 
$$ (\omega, v) = \sum_i \omega_i v^i. $$


If $f: V \to W$ is a linear map, then there is a dual map $f^*: V^* \to W^*$, given by 
$$ f(\varphi)(w) = \varphi(f(w)), \quad w \in W, \varphi \in V^*. $$


Now, let $M$ be a smooth manifold, $p \in M$, $T_p M$ its tangent space over $p$. We define the cotangent space at $p$ to be  $$T^*_p M: = (T_p M)^*.$$

Given a coordinate system near $p$, $(U, (x^1, \cdots, x^n))$, we have basis for $T_p M$ as $\{ \d_i|_p = \frac{\d}{\d x^i}\vert_p$. The dual basis of $\{\d_i\}|_p$ is denoted as $\{dx^i\}|_p$. Hence, a covector $\omega$ at $p$ can be written as 
$\omega = \sum_i \omega_i dx^i|_p$

Exercise: figure out the transformation rule for the coefficients $\omega_I$ if we change coordinates. 

==== Covector Fields (Differential 1-form) ====
Recall how we define smooth vector field over $M$: it is an assignment $p \mapsto X(p) \in T_p M$ for $p \in M$, such that if we write it in coordinate patch $(U, (x_i))$, we have
$$ X(p) = \sum_i X(p)^i \d_i $$
where $X(p)^i:U \to \R$ are smooth functions. 

Simiarly, we define smooth covector fields over $M$ as an assignment $p \mapsto \omega(p) \in T^*_p M$, such that in coordinates we have 
$$ \omega(p) = \sum_i \omega(p)_i dx^i $$
where $\omega(p)_i$ are smooth. 

** Prop **: The cotangent bundle $T^*M = \sqcup_p T_p^*M $ is a vector bundle over $M$.
Lecture Notes

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