---- - $B$ is [[smooth manifold]] of dimension $n$. - $E \xrightarrow{\pi} B$ is a rank-$m$ smooth [[vector bundle]] with typical fiber $V$. $B$ is a [[smooth manifold]]. - Recall that the notations $\Gamma(E)=\Omega^{0}_{B}(E)$ and $\Gamma(T^{*}B \otimes E)=\Omega_{B}^{1}(E)$ denote the spaces of sections and [[differential form with values in a vector bundle|vector valued 1-forms]] respectively. > [!definition] Definition. ([[parallel section]]) > Let $\nabla^{}:\Omega^{0}_{B}(E) \to \Omega^{1}_{B}(E)$ be a [[covariant derivative on a vector bundle|covariant derivative]] on $E$. Let $\gamma:I \to B$ be a [[parameterized curve]] in $B$. A section $s$ for which $\nabla s=0$ is called a **parallel section**. > A [[pullback of a vector bundle|section]] $s_{\gamma} \in \Gamma(\gamma^{*}E)$ (so $s_{\gamma}(t) \in E_{\gamma(t)}$) is called a **parallel section along $\gamma$** if $\nabla^{E}_{\dot{\gamma}} s_{\gamma}=0$, where the [[covariant derivative on a vector bundle|covariant derivative]] is understood to be taken via the [[pullback connection]]. ^definition [[partial covariant derivative|So]] - $s_{\gamma} \in \gamma^{*}E$ is called a **parallel section along $\gamma$ if $(\gamma^{*}\nabla)_{\frac{ \partial }{ \partial t }}(s_{\gamma})=0$. - $s \in \Gamma(E)$ is called a **parallel section along $\gamma$** if its [[pullback of a vector bundle|pullback]] $s_{\gamma}:=\gamma^{*}s$ is. Observing $(\gamma^{*}\nabla)_{\frac{ \partial }{ \partial t }} (\gamma^{*}s)=\gamma^{*}\left( \nabla_{\gamma _{*} \frac{ \partial }{ \partial t } } s \right ) =(\nabla_{\gamma_{*} \frac{ \partial }{ \partial t }} s \circ \gamma, \id) , $ and recalling the various notations $\gamma_{*} \frac{ \partial }{ \partial t } =d \gamma_{t} \left( \frac{ \partial }{ \partial t } |_{t}\right)=\dot{\gamma}(t) \in T_{\gamma(t)}M$, we have that $s$ is parallel along $\gamma$ if and only if $\nabla_{\dot{\gamma}} s \circ \gamma=0.$ In [[coordinate chart|local coordinates]]/trivialization $\big( U, ((x^{k})_{k=1}^{n}, (a^{j})_{j=1}^{m} )\big)$ , $\nabla^{}=d+A$ for some [[connection on a vector bundle|connection 1-form]] $A=(\Gamma^{i}_{jk}) \in \Omega^{1}_{U}(\operatorname{End }E)$ and we see the requirement is $\begin{align} \big(\overbrace{ (\frac{ \partial s _{\gamma}^{i} }{ \partial x^{k} } + \Gamma^{i}_{jk}s_{\gamma} ^{j})dx^{k} }^{ (d+ A)s_{\gamma} } \big) \, (\overbrace{ \dot{x}^{k} \frac{ \partial }{ \partial x^{k} } }^{ \dot{\gamma} })=0 \end{align}$ for all $i \in [m]$. Performing the evaluation, we obtain the [[linear system of ODEs|linear]] [[ODE]] [[system of ODEs|system]] ($i=1,\dots,m$)[^1] $\frac{ds_{\gamma}^{i}}{dt}(t)+ \Gamma^{i}_{jk}\big( \gamma(t) \big) \, \dot{x}^{k}(t) s_{\gamma}^{j}(t)=0.$ Given an initial condition $s_{\gamma}(t_{0})=v \in E_{\gamma(t_{0})}$, there [[Existence Theorem for Linear Systems of ODEs|exists]] a [[uniqueness of ODE solution|unique]] solution $s_{\gamma}$; the resulting [[linear isomorphism]] $P_{t_{0} \to t} : E_{\gamma(t_{0})} \to E_{\gamma(t)}, \, \, v \mapsto s_{\gamma}(t),$ is called the **parallel transport from $t_{0}$ to $t$ along $\gamma$**. [[horizontal lift with respect to a connection on a vector bundle|horizontal lift]] [^1]: Here we have used the [[chain rule]] $\frac{ \partial s_{\gamma}^{i} }{ \partial x^{k} } |_{\gamma(t)} \dot{x}^{k}(t)=\frac{d}{dt}(s ^{i} \circ \gamma)(t)=\frac{d s ^{i}_{\gamma }}{dt} (t)$. I think I want to come back and be a little more rigorous here. > [!equivalence] > ^equivalence ---- #### ---- #### 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 > ```