"connection *connects* tangent spaces" https://www.youtube.com/watch?v=MRU2D6sLpU0 <-genuinely nice ---- - $B$ 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 $s_{\gamma}$ ([[integral curve]] for [[vector field|vector fields]]); 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] > For notation with the [[Levi-Civita connection]] specifically: > The goal is to obtain a [[vector field]] $X$ along $\gamma$ "with constant velocity", in the sense that $\nabla_{\dot{\gamma}}X=0.$ We call such $X$ **parallel along $\gamma$**. In [[coordinate chart|local coordinates]] $(x^{i})$, [[partial covariant derivative|this is]] asserting that for all $i=1,\dots,n$, $\Gamma^{i}_{jk} (\gamma(t))\, \dot{\gamma}^{j}(t)X^{k}(\gamma(t)) + \frac{ \partial X^{i} }{ \partial x^{k} }(\gamma(t))\, \dot{\gamma}^{k}(t) =0 \text{ for all } t \in I.$ Undoing the [[chain rule]] (and suppressing precomposition with $\gamma$) this is $\Gamma^{i}_{jk}(t)\dot{\gamma}^{j}(t)X^{k}(t)+ \frac{dX^{i}}{dt}=0.$ ^equivalence Isometry: $\frac{d}{dt} g(\tau_{t}X_{p}, \tau_{t}X_{p})=2g(\underbrace{ \nabla_{\dot{\alpha}}X }_{ =0 }, X)=0$. ---- #### the [[integral curve|flow]][^5] $\tau:I \times \gamma^{*}TM \to \gamma^{*}TM$ of this [[ODE]] is called **parallel transport along $\gamma$**. [^5]: By 'flow' here we just mean the map $\tau$ which takes in a time $t$ and some $X_{p} \in T_{p}M$ with $p \in \operatorname{im } \gamma$ and outputs $X(t)$, the solution to the ODE with initial condition $X_{p}$. ---- #### 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 > ```