- Let $E \xrightarrow{\pi}B$ be a [[vector bundle|smooth vector bundle]] with typical fiber $\mathbb{R}^{m}$ over a [[smooth manifold]] $B$ of dimension $n$. - Let $U \subset B$ simultaneously a denote a [[coordinate chart|coordinate neighborhood]] on $B$ and a [[vector bundle|trivializing neighborhood]] for $E$. - Fix coordinates $( x^{k})=(x^{1},\dots,x^{n})$ on $U$, and $( a^{j})=(a^{1},\dots,a^{m})$ on the typical fiber $\mathbb{R}^{m}$. Summation convention is in effect. > [!definition] Definition. ([[connection on a vector bundle]]) > Recall the notion of a [[field of horizontal subspaces]] $S=\{ S_{p}: p \in E \}$; in particular that any [[vector bundle|local trivialization]] $E_{U} \xrightarrow{\Phi_{U}}U \times \mathbb{R}^{m}$ uniquely determines an expression $S_{p}=\bigcap_{i=1}^{m} \text{ker }\theta^{i}_{p} \text{ with } \theta^{i}_{p}=da^{i}+e_{k}^{i}(\overbrace{x,a}^{p}) \ dx^{k} \text{ and } e_{k}^{i} \in C^{\infty}(U \times \mathbb{R}^{m})$ for each $p \in E$, ($p=(x,a)$ in the local trivialization). We call $S$ a **connection on $E$** if the functions $e_{k}^{i}$ are *linear* in the fiber variables[^8] (latter argument; in $a$), i.e., if for any fixed $x \in U$ $e_{k}^{i}(x,-)$ may be expressed $e_{k}^{i}(x, -)=\Gamma_{jk}^{i}(x) \ a^{j} $ and so $\theta_{p}^{i}=da^{i} + \Gamma_{jk}^{i}(x) \ a^{j} \ dx^{k}=da^{i} + A_{j}^{i}(x) a^{j}$ where $\Gamma_{jk}^{i}:U \subset B \to \mathbb{R}$ are smooth functions called **coefficients of the connection $S_{p}$ in the local trivialization $\Phi_{U}$**. $A_{j}^{i} \in \Omega^{1}(U)$ is a [[differential form|differential]] $1$-[[differential form|form]] on $U \subset B$, $A_{j}^{i}=\Gamma^{i}_{jk} \ dx^{k}$; $A^{i}_{j}(x)=\Gamma^{i}_{jk}(x) \ dx^{k}$. > So locally a connection looks like a 'matrix $A=(A^{i}_{j})_{i,j=1}^{m}$ of differential 1-forms', $A |_{U} \in \Omega_{U \subset B}(\text{End }E)$. Or we could call it a 'matrix-valued differential 1-form', since $x \in U$ determines a matrix of 1-forms $A(x)$. *Globally* a connection is not a differential form-like thing, however, because it follows a different transformation law. > $A$ is called the **connection $1$-form**, and is sometimes used to denote the connection itself. > [!proposition] The Transformation Law for a Connection. > > **Setup.** > Suppose $\Phi'=\Phi_{U'}'$ is another local trivialization over some $U'$ with $U \cap U' \neq \emptyset$. Have transition function $\psi=(\psi^{i'}_{i}) \in C^{\infty}(U \cap U'; \mathbb{R}^{m \times m})$ from $\Phi$ to $\Phi'$. Its inverse is the transition from $\Phi'$ back to $\Phi$: $\psi ^{-1}=(\psi^{i}_{i'})$. And $a^{i'}=\psi^{i'}_{i}a^{i}$. > > **Procedure.** In one big step: > > $\begin{align} > \theta^{i}&= \psi^{i}_{i'} \textcolor{Thistle}{\theta^{i'}} \\ > &= \psi^{i}_{i'} (\textcolor{Thistle}{\underbrace{ da^{i'} }_{ =d(\psi^{i}_{i'}a^{i}) }+A^{i'}_{j'}a^{j'}}) \\ > &= \psi^{i}_{i'} ( \overbrace{ \textcolor{Skyblue}{d(\psi^{i'}_{i} a^{i})} }^{ } + A^{i'}_{j'} a^{j'} ) \\ > &= \psi^{i}_{i'} \big( \textcolor{Skyblue}{\underbrace{ (d \psi^{i'}_{i} )a^{i} }_{ } + \psi^{i'}_{i} da^{i}} + A^{i'}_{j'} a^{j'}\big) \\ > &= \textcolor{Skyblue}{da^{i}} + \psi^{i}_{i'} \textcolor{Skyblue}{\overbrace{ d \psi^{i'}_{j}a^{j} }^{ \text{m-v mult.} }} + \psi^{i}_{i'} A^{i'}_{j'} \underbrace{ a^{j'} }_{ \psi^{j'}_{j} a^{j} } \\ > &= da^{i} + \underbrace{ (\psi^{i}_{i'} d\psi^{i'}_{j} + \psi^{i}_{i'} A^{i'}_{j'} \psi^{j'}_{j}) }_{ A^{i}_{j} } a ^{j}. > \end{align}$ > So the [[transformation law]] is $A^{i}_j=\psi^{i}_{i'} d \psi^{i}_{j} + \psi^{i}_{i'} A^{i'}_{j'} \psi^{j'}_{j},$ > or in matrix form: $A=\psi d \psi ^{-1} + \psi A'\psi ^{-1}=\psi A'\psi ^{-1}-(d \psi) \psi ^{-1}.$ A^{\psi \Phi}=\psi A^{\Phi}\psi ^{-1} + \psi d \psi ^{-1} = \psi A ^{\Phi}\psi ^{-1}- (d \psi) \psi ^{-1} [^8]: That is, for fixed $x \in B$ the map $e^{i}_{k}(x, -):\mathbb{R}^{m} \to \mathbb{R}$ belongs to $(\mathbb{R}^{m})^{*}$, i.e., is [[linear map|linear]]. Of course, $e^{i}_{k}(x, -) \in (\mathbb{R}^{m})^{*}$ if and only if $e^{i}_{k}(x,-)$ can be written as a [[linear combination]] $e^{i}_{k}(x, -)= \Gamma^{i}_{jk}(x) \, a^{j}$ of the coordinate basis functionals $a^{j}$. ---- #### Now [[exterior derivative|exterior-differentiate]] $a^{i'}$ thus: $d(a^{i'})=d(\psi^{i'}_{i} a^{i})=(d \psi^{i'}_{i} )a^{i} + \psi^{i'}_{i} da^{i}.$ We have $\theta^{i'}=da^{i'}+A^{i'}_{j'}a^{j'}$, and thus $\theta^{i'}=(d \psi^{i'}_{i} )a^{i} + \psi^{i'}_{i} da^{i} + A^{i'}_{j'}a^{j'}$ and ---- #### 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 > ``` (may/may not be useful: https://en.wikipedia.org/wiki/Local_system)