---- > [!definition] Definition. ([[horizontal lift with respect to a connection on a vector bundle]]) > Let $E \xrightarrow{\pi}B$ be a smooth [[vector bundle]] on a [[smooth manifold]] $B$, endowed [[connection on a manifold|with]] [[connection on a vector bundle|connection]] $A=(\Gamma^{i}_{jk})$, where $i,j=1,\dots,\text{rank }E$ and $k=1,\dots,\text{dim }B$. > > A [[smooth maps between manifolds|smooth]] [[parameterized curve]] $\gamma(t):I \to B$ on the base $B$ may be written in [[coordinate chart|local coordinates]] as $\big( x^{k}(t) \big)$. A [[lifting|lift]] $\gamma_{E}(t)$ of this $\gamma$ to $E$ is locally expressed as $\big( x^{k}(t), a^{j}(t) \big)$, using local trivialization of the bundle $E$ to define coordinates $a^{j}$ along the fibers. > > We can ask for $\gamma_{E}$ to be a **horizontal lift**, in the sense that, for all $t$, the [[tangent vector to a smooth manifold|velocity vector]] $\dot{\gamma}_{E}(t) \in T_{\gamma_{E}(t)}E$ lives in a [[linear subspace|subspace]] $S_{\gamma_{E}(t)}$ that is [[horizontal subspace|horizontal]] with respect to the [[connection on a vector bundle|connection]] $A$. Explicitly, [[connection on a vector bundle|this is saying that]] $\begin{align} > \theta^{i}\big(\dot{\gamma}_{E}(t) \big)&=0 \text{ for all }i =1,\dots,\text{rank }E, \text{ where }\\ > \theta^{i} : T^{*} _{\gamma_{E}(t)}E& \to \mathbb{R} \text{ is the linear functional} \\ > \theta^{i} & = da^{i} + \Gamma^{i}_{jk}(x) \ a^{j} dx^{k} > \end{align}$ > for all $t$. In other words, a horizontal lift of the curve $\gamma$ is a solution to the [[linear system of ODEs|linear system of]] [[ODE|ODEs]] for $\big(a^{j}(t)\big)_{j=1}^{\text{rank }E}$[^1] $\begin{align} > 0=\dot{a}^{i} + \Gamma^{i}_{jk}(x) \ a^{j} \dot{x}^{k} \tag{{\dagger}} > \end{align}$ > on the [[interval]] $I$. According to standard linear ODE theory, $\ref{{\dagger}}$ is [[Existence Theorem for Linear Systems of ODEs|solvable]] and the solution is [[Uniqueness for System of Linear ODEs|unique]] given initial condition $\big( a^{j}(0) \big)_{j=1,\dots,\text{rank }E}$ coming from $\gamma_{E}(0) \in \pi ^{-1}\big( \gamma(0) \big) \in E$. > ![[Pasted image 20250518103932.png]] > [!equivalence] > ^equivalence Examining [[covariant derivative along a curve|this discussion]] tells us that a lift $\gamma_{E}$ is horizontal if and only if $\nabla^{E}_{\dot{\gamma}_{E}} \dot{\gamma}_{E}=0$, [^1]: The coordinate expression for $\dot{\gamma}_{E}(t)$ is $\dot{x}^{k} \frac{ \partial }{ \partial x^{k} }+\dot{a}^{j} \frac{ \partial }{ \partial a^{j} }$; plugging this into the horizontal subspace equation gives $\begin{align} da^{i}(&\dot{x}^{k} \frac{ \partial }{ \partial x^{k} }+\dot{a}^{j} \frac{ \partial }{ \partial a^{j} })+ \Gamma^{i}_{jk} a^{j} dx^{k}(\dot{x}^{k} \frac{ \partial }{ \partial x^{k} }+\dot{a}^{j} \frac{ \partial }{ \partial a^{j} }) \\ &= \dot{a}^{i} + \Gamma^{i}_{jk} a^{j} \dot{x}^{k} \end{align}$ ---- #### ---- #### 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 > ```