---- > [!definition] Definition. ([[vector bundle]]) > Let $M$ be a [[topological space]]. A **(real) vector bundle** of rank $k$ over $M$ is a [[topological space]] $E$ together with a [[continuous]] [[surjection]] $\pi: E \to M$ satisfying: > >**1.** For each $p \in M$, the fiber $E_{p}:= \pi ^{-1}(p)$ over $p$ is endowed with the structure of a $k$-dimensional $\mathbb{R}$-[[vector space]] > **2.** For each $p \in M$, there exist a [[neighborhood]] $U$ of $p$ in $M$ and a [[homeomorphism]] $\Phi:\pi ^{-1}(U) \to U \times \mathbb{R}^{n}$ (called a **local trivialization of $E$ over $U$**) such that >1. $\pi_{U} \circ \Phi=\pi$ (where $\pi_{U}:U \times \mathbb{R}^{k} \to U$ is the [[product topology|projection]]) >2. For each $q \in U$, the restriction $\Phi |_{E_{q}}$ is a [[linear isomorphism|isomorphism of vector spaces]] $E_{q} \to \{ q \} \times \mathbb{R}^{k}$. $E$ is called the **total space of the bundle**, $M$ is called its **base**, and $\pi$ its **projection**. > > If $M,E$ are in fact [[smooth manifold|smooth manifolds]], $\pi$ is a [[smooth maps between manifolds|smooth map]], and the local trivializations $\Phi$ can be chosen to [[diffeomorphism|diffeomorphisms]], then $E$ is again a [[smooth manifold]][^2] and $\Phi$ a **smooth local trivialization**. In this case $\pi$ turns out to be a [[smooth submersion]]. > A rank-$1$ vector bundle is often called a **line bundle**. ^definition [^2]: Let $p \in E$, $b:=\pi(p)$. Let $(W, \varphi:W \to \mathbb{R}^{n})$ be a [[coordinate chart|chart]] around $b$. Let $(U, \Phi_{U}: \pi ^{-1}(U) \to U \times \mathbb{R}^{k})$ be a [[vector bundle|local trivialization]] about $b$. Then with $V:= W \cap U$, we define a [[coordinate chart|chart]] about $p$ as $\pi ^{-1} (V) \xrightarrow{\Phi |_{V}} V \times \mathbb{R}^{k} \xrightarrow{\varphi |_{V} \times \id} \mathbb{R}^{n} \times \mathbb{R}^{k}.$ In general, we (often implicitly) assume $U=W$, so that coordinate neighborhoods are also trivializing neighborhoods. > [!definition] Definition. (Sections) > A **(smooth) (global) section** of $E$ is a (smooth) map $s:M \to E$ satisfying $\pi \circ s=\id_{B}$. > A **(smooth) local section** is a (smooth) map $s:N \to E$, $N$ open in $M$, such that $\pi \circ s=\id_{N}$. > > > ![[CleanShot 2025-03-25 at [email protected]]] > Image: zero section of the [[Mobius bundle]] (https://commons.wikimedia.org/wiki/File:Universal_line_bundle.jpg). > > Sections are a way to think about some suitably compatible (e.g. [[continuous]], smooth, etc.) assignment of each $p \in M$ to an element of the fiber $E_{p}$. Indeed, a map $s:M \to E$ satisfies $s(p) \in E_{p}$ for all $p$ iff $s$ is a section.[^1] In such a sense, sections of a vector bundle generalize vector-valued functions on open domains in $\mathbb{R}^{n}$. > A [[vector bundle|local trivialization]] $(U, \Phi)$ determines a collection of smooth local sections $\big(e_{i}: U \to \pi ^{-1}(U)\big)_{i=1}^{\text{rank }E}$ such that $\big(e_{1}(b), \dots, e_{\text{rank }E}(b)\big)$ forms a [[basis]] of $E_{b}$ for all $b \in U$.[^4] We call $(e_{i})$ the **(smooth) local frame over $U$ determined by $\Phi$**. [^4]: Indeed, $\Phi |_{E_{b}}$ determines a [[linear isomorphism|linear]] [[module is free iff admits basis|coordinate isomorphism]] $E_{b} \xleftarrow{\sim} \mathbb{R}^{m}$, which is [[module is free iff admits basis|by definition]] of choice of [[basis]] for $E_{b}$. This choice varies smoothly because $\Phi$ is smooth. > [!definition] Definition. (Zero section) > The **zero section** of a [[vector bundle]] is $s_{0}:X\to E$ given by $s_{0}(x)=0 \in E_{x}$. A few topological facts: > - All [[section|sections]] are [[homotopy|homotopic]] to $s_{0}$, hence to each other. > - The composition $E \xrightarrow{\pi}X \xrightarrow{s_{0}}E$ > is [[homotopy|homotopic]] to the [[identity map]] $\id_{E}$, such each fiber $E_{x}$ is [[contractible]]. Hence $s_{0}$ is a [[homotopy equivalent|homotopy equivalence]] $X \to E$. Since all sections are [[homotopy|homotopic]] to $s_{0}$, *every* section is a [[homotopy equivalent|homotopy equivalence]] $X \to E$. > > > - [ ] short argument in notes justifying these > > ^definition > [!basicexample] > - [[product bundle]] > - [[mobius bundle]] > - [[tangent bundle]] > - [[cotangent bundle]] > - [[dual vector bundle]] > - [[exterior bundle]]? > - [[tautological vector bundle]] ^basic-example ---- #### [^1]: To see this: $s(p) \in E_{p}=\pi ^{-1}(p)$ for all $p$ implies $\pi(s(p))=p$. Conversely, if $s$ is a section, then $\pi\big( s(p) \big)=p$ for all $p \in M$, meaning $s(p) \in \pi ^{-1}(p)=E_{p}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```