# Differentiable Manifolds/Vector (sub-)bundles, sections, foliations, distributions and Frobenius' theorem
## What are vector bundles?

Definition 10.1:
Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in N_{0}\cup \{\infty \}$ as usual. A vector bundle of $M$ is a manifold $E$ of class ${\mathcal {C}}^{n}$ together with
* a function $\pi _{E}:E\to M$ (which we shall call projection of $E$) such that for each $p\in M$, $\pi _{E}^{-1}(p)$ is a finite-dimensional vector space over the real numbers AND
* for each $p\in M$ an open set $O\subseteq M$ with $p\in O$ and such that there is a diffeomorphism $\psi _{E,O}:O\times \mathbb {R} ^{k_{p}}\to E$ of the product manifold $M\times \mathbb {R} ^{k_{p}}$ to the manifold $E$ of class ${\mathcal {C}}^{n}$ (which shall be called bundle chart of $E$), where $k_{p}\in \mathbb {N} _{0}$ is the dimension of $\pi ^{-1}(p)$, such that for each $p\in O$ the function $\psi _{E}(p,\cdot ):\mathbb {R} ^{k_{p}}\to E$ has $\pi _{E}^{-1}(p)$ as its image and is a linear isomorphism.
Lemma 10.2: Let $E$ be a vector bundle of the manifold $M$ with projection $\pi _{E}$. Then if $p,q\in M$ are contained in the same connected component of $M$, the dimensions of $\pi _{E}^{-1}(p)$ and $\pi _{E}^{-1}(q)$ are equal.
Proof:
We define the function $\mu :M\to \mathbb {N} _{0},\mu (q):={\text{dim }}\pi _{E}^{-1}(q)$. Let now $p\in M$ be contained in the connected component of $M$ $Q$. If we pick any point $q\in M$, due to the definition of a vector bundle, there is an open set $O\subseteq M$ with $q\in O$ and a bundle chart $\psi _{E,O}:O\times \mathbb {R} ^{k_{q}}$, where $k_{q}$ is the dimension of $$, such that $\psi _{E}(p,\cdot ):\mathbb {R} ^{k_{q}}\to E$ has $\pi _{E}^{-1}(p)$ as its image and is a linear isomorphism. Therefore, for all $r\in O$, the dimension of $\pi _{E}^{-1}(r)$ is equal to $k_{q}$.
From this follows that the set $\{q\in M|\mu (q)=\mu (p)\}$ and its complement in $M$ are both open (since for every point $q$ in the complement of this set there also exists an open neighbourhood such that all points $r$ in this neighbourhood satisfy ${\text{dim }}\pi _{E}^{-1}(r)={\text{dim }}\pi _{E}^{-1}(q)$ and thus all the points $r$ in that neighbourhood are also contained in the complement).
But if the set $\{q\in M|\mu (q)=\mu (p)\}$ and its complement in $M$ are both open, then so are the respective intersections with $Q$ with respect to the subspace topology on $Q$ induced by the topology of $M$. But, by definition of a connected component, $Q$ is a connected set with respect to the subspace topology, and thus, since $\{q\in M|\mu (q)=\mu (p)\}\cap Q$ is open and closed (since the complement is open) and nonempty (as it contains $p$), this set equals the whole set $Q$ by definition of connected sets.$\Box$

## The tangent and cotangent bundles as part of a respective vector bundle

In the following section we want to show, that both tangent and cotangent bundle with a specific atlas are manifolds, and if we define specific projections and bundle charts, we obtain vector bundles out of them and the tangent and cotangent bundles.
Definition 10.3:
Let $M$ be a manifold and let $O\subseteq M$ be an open subset of $M$. Then we define:
    $TO:=\bigcup _{q\in O}T_{q}M$
Theorem 10.4:
Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, where $n\in N_{0}\cup \{\infty \}$, with atlas $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$. For each $\upsilon \in \Upsilon$, we define the function
    $\theta _{\upsilon }:TO_{\upsilon }\to \mathbb {R} ^{2d},\theta _{\upsilon }\left(\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial {\phi _{\upsilon }}_{j}}}\right)_{q}\right):=(\phi _{\upsilon }(q),a_{1},\ldots ,a_{d})$
and the set
    $U_{\upsilon }:=\{(q,T_{q}M)|q\in O_{\upsilon }\}$
Further, we define the topology on $TM$ to be
    $\left\{\theta _{\upsilon }^{-1}(V)|V{\text{ open in }}\mathbb {R} ^{2d},\upsilon \in \Upsilon \right\}$
This is a topology and $\{(U_{\upsilon },\theta _{\upsilon })|\upsilon \in \Upsilon \}$ is an atlas of class ${\mathcal {C}}^{n}$ and $TM$ together with $\{(U_{\upsilon },\theta _{\upsilon })|\upsilon \in \Upsilon \}$ is a $2d$-dimensional manifold of class ${\mathcal {C}}^{n}$.
Theorem 10.5:
Let $TM$ be equipped with the atlas as defined in the statement of theorem 10.4. If we define
    $\pi :TM\to M,\pi (\mathbf {V} _{p}):=p$
and for each $\upsilon \in \Upsilon$
    $\psi _{TM,O_{\upsilon }}:O_{\upsilon }\times \mathbb {R} ^{d}\to TM,\psi _{TM,O_{\upsilon }}(p,a_{1},\ldots ,a_{d}):=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial {\phi _{\upsilon }}_{j}}}\right)_{p}$,
then $TM$ with $\pi$ as projection and the functions $\psi _{TM,O_{\upsilon }}$ is a vector bundle.
Definition 10.6:
Let $M$ be a manifold and let $O\subseteq M$ be an open subset of $M$. Then we define:
    $TO^{*}:=\bigcup _{q\in O}T_{q}M^{*}$
Theorem 10.7:
Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$.