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

 Differentiable Manifolds ← Product manifolds and Lie groups Vector (sub-)bundles, sections, foliations, distributions and Frobenius' theorem Differential operators and curvature →

## What are vector bundles?

Definition 10.1:

Let ${\displaystyle M}$ be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$, where ${\displaystyle n\in N_{0}\cup \{\infty \}}$ as usual. A vector bundle of ${\displaystyle M}$ is a manifold ${\displaystyle E}$ of class ${\displaystyle {\mathcal {C}}^{n}}$ together with

• a function ${\displaystyle \pi _{E}:E\to M}$ (which we shall call projection of ${\displaystyle E}$) such that for each ${\displaystyle p\in M}$, ${\displaystyle \pi _{E}^{-1}(p)}$ is a finite-dimensional vector space over the real numbers AND
• for each ${\displaystyle p\in M}$ an open set ${\displaystyle O\subseteq M}$ with ${\displaystyle p\in O}$ and such that there is a diffeomorphism ${\displaystyle \psi _{E,O}:O\times \mathbb {R} ^{k_{p}}\to E}$ of the product manifold ${\displaystyle M\times \mathbb {R} ^{k_{p}}}$ to the manifold ${\displaystyle E}$ of class ${\displaystyle {\mathcal {C}}^{n}}$ (which shall be called bundle chart of ${\displaystyle E}$), where ${\displaystyle k_{p}\in \mathbb {N} _{0}}$ is the dimension of ${\displaystyle \pi ^{-1}(p)}$, such that for each ${\displaystyle p\in O}$ the function ${\displaystyle \psi _{E}(p,\cdot ):\mathbb {R} ^{k_{p}}\to E}$ has ${\displaystyle \pi _{E}^{-1}(p)}$ as its image and is a linear isomorphism.

Lemma 10.2: Let ${\displaystyle E}$ be a vector bundle of the manifold ${\displaystyle M}$ with projection ${\displaystyle \pi _{E}}$. Then if ${\displaystyle p,q\in M}$ are contained in the same connected component of ${\displaystyle M}$, the dimensions of ${\displaystyle \pi _{E}^{-1}(p)}$ and ${\displaystyle \pi _{E}^{-1}(q)}$ are equal.

Proof:

We define the function ${\displaystyle \mu :M\to \mathbb {N} _{0},\mu (q):={\text{dim }}\pi _{E}^{-1}(q)}$. Let now ${\displaystyle p\in M}$ be contained in the connected component of ${\displaystyle M}$ ${\displaystyle Q}$. If we pick any point ${\displaystyle q\in M}$, due to the definition of a vector bundle, there is an open set ${\displaystyle O\subseteq M}$ with ${\displaystyle q\in O}$ and a bundle chart ${\displaystyle \psi _{E,O}:O\times \mathbb {R} ^{k_{q}}}$, where ${\displaystyle k_{q}}$ is the dimension of ${\displaystyle }$, such that ${\displaystyle \psi _{E}(p,\cdot ):\mathbb {R} ^{k_{q}}\to E}$ has ${\displaystyle \pi _{E}^{-1}(p)}$ as its image and is a linear isomorphism. Therefore, for all ${\displaystyle r\in O}$, the dimension of ${\displaystyle \pi _{E}^{-1}(r)}$ is equal to ${\displaystyle k_{q}}$.

From this follows that the set ${\displaystyle \{q\in M|\mu (q)=\mu (p)\}}$ and its complement in ${\displaystyle M}$ are both open (since for every point ${\displaystyle q}$ in the complement of this set there also exists an open neighbourhood such that all points ${\displaystyle r}$ in this neighbourhood satisfy ${\displaystyle {\text{dim }}\pi _{E}^{-1}(r)={\text{dim }}\pi _{E}^{-1}(q)}$ and thus all the points ${\displaystyle r}$ in that neighbourhood are also contained in the complement).

But if the set ${\displaystyle \{q\in M|\mu (q)=\mu (p)\}}$ and its complement in ${\displaystyle M}$ are both open, then so are the respective intersections with ${\displaystyle Q}$ with respect to the subspace topology on ${\displaystyle Q}$ induced by the topology of ${\displaystyle M}$. But, by definition of a connected component, ${\displaystyle Q}$ is a connected set with respect to the subspace topology, and thus, since ${\displaystyle \{q\in M|\mu (q)=\mu (p)\}\cap Q}$ is open and closed (since the complement is open) and nonempty (as it contains ${\displaystyle p}$), this set equals the whole set ${\displaystyle Q}$ by definition of connected sets.${\displaystyle \Box }$

## 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 ${\displaystyle M}$ be a manifold and let ${\displaystyle O\subseteq M}$ be an open subset of ${\displaystyle M}$. Then we define:

${\displaystyle TO:=\bigcup _{q\in O}T_{q}M}$

Theorem 10.4:

Let ${\displaystyle M}$ be a ${\displaystyle d}$-dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$, where ${\displaystyle n\in N_{0}\cup \{\infty \}}$, with atlas ${\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}$. For each ${\displaystyle \upsilon \in \Upsilon }$, we define the function

${\displaystyle \theta _{\upsilon }:TO_{\upsilon }\to \mathbb {R} ^{2d},\theta _{\upsilon }\left(\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial {\phi _{\upsilon }}_{j}}}\right)_{q}\right):=(\phi _{\upsilon }(q),a_{1},\ldots ,a_{d})}$

and the set

${\displaystyle U_{\upsilon }:=\{(q,T_{q}M)|q\in O_{\upsilon }\}}$

Further, we define the topology on ${\displaystyle TM}$ to be

${\displaystyle \left\{\theta _{\upsilon }^{-1}(V)|V{\text{ open in }}\mathbb {R} ^{2d},\upsilon \in \Upsilon \right\}}$

This is a topology and ${\displaystyle \{(U_{\upsilon },\theta _{\upsilon })|\upsilon \in \Upsilon \}}$ is an atlas of class ${\displaystyle {\mathcal {C}}^{n}}$ and ${\displaystyle TM}$ together with ${\displaystyle \{(U_{\upsilon },\theta _{\upsilon })|\upsilon \in \Upsilon \}}$ is a ${\displaystyle 2d}$-dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$.

Theorem 10.5:

Let ${\displaystyle TM}$ be equipped with the atlas as defined in the statement of theorem 10.4. If we define

${\displaystyle \pi :TM\to M,\pi (\mathbf {V} _{p}):=p}$

and for each ${\displaystyle \upsilon \in \Upsilon }$

${\displaystyle \psi _{TM,O_{\upsilon }}:O_{\upsilon }\times \mathbb {R} ^{d}\to TM,\psi _{TM,O_{\upsilon }}(p,a_{1},\ldots ,a_{d}):=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial {\phi _{\upsilon }}_{j}}}\right)_{p}}$,

then ${\displaystyle TM}$ with ${\displaystyle \pi }$ as projection and the functions ${\displaystyle \psi _{TM,O_{\upsilon }}}$ is a vector bundle.

Definition 10.6:

Let ${\displaystyle M}$ be a manifold and let ${\displaystyle O\subseteq M}$ be an open subset of ${\displaystyle M}$. Then we define:

${\displaystyle TO^{*}:=\bigcup _{q\in O}T_{q}M^{*}}$

Theorem 10.7:

Let ${\displaystyle M}$ be a ${\displaystyle d}$-dimensional manifold of class ${\displaystyle {\mathcal {C}}^{n}}$.

## Foliations, distributions and Frobenius' theorem

 Differentiable Manifolds ← Product manifolds and Lie groups Vector (sub-)bundles, sections, foliations, distributions and Frobenius' theorem Differential operators and curvature →