Parallel section along a curve

Let $\pi:E\to M$ be a smooth vector bundle over a smooth manifold $M$ , equipped with a connection on a vector bundle $\nabla$ .

Let $I\subset \mathbb{R}$ be an interval and $\gamma:I\to M$ a smooth curve. A section of $E$ along $\gamma$ is a smooth section of the pullback bundle $\gamma^*E$ , equivalently a smooth map $s:I\to E$ such that $\pi\circ s=\gamma$ . The connection $\nabla$ induces a pullback connection on $\gamma^*E$ , and hence a covariant derivative $Ds/dt$ along the curve.

A section $s$ along $\gamma$ is called parallel along $\gamma$ if its covariant derivative for the pullback connection vanishes:

\frac{D s}{dt}(t)=0 \quad \text{for all } t\in I.

Existence and uniqueness

Given $t_0\in I$ and $v_0\in E_{\gamma(t_0)}$ , there is a unique parallel section $s$ along $\gamma$ with $s(t_0)=v_0$ . The resulting identification of fibers is the parallel transport determined by $\nabla$ along $\gamma$ .

Examples

Trivial bundle over an interval. Let $E=I\times \mathbb{R}^n\to I$ with the standard (componentwise) connection. A section along the identity curve $\gamma(t)=t$ is a map $s(t)=(t,v(t))$ . The parallel condition is $v'(t)=0$ , so parallel sections are exactly the constant vectors $v(t)\equiv v_0$ .

Tangent bundle of Euclidean space. Take $E= T\mathbb{R}^m$ with the standard flat connection. Along any smooth curve $\gamma$ , a vector field $V(t)\in T_{\gamma(t)}\mathbb{R}^m\cong \mathbb{R}^m$ is parallel iff its coordinate vector in $\mathbb{R}^m$ is constant in $t$ .

A rank-one connection written as an ODE. On a trivial real line bundle $E=M\times \mathbb{R}$ , any connection can be written locally as $\nabla=d+\alpha$ for a 1-form $\alpha$ on $M$ . Along $\gamma$ , a section is a function $f(t)$ , and the parallel condition becomes $f'(t) + \alpha(\dot\gamma(t))\,f(t)=0,$ so $f(t)=f(t_0)\exp\!\left(-\int_{t_0}^t \alpha(\dot\gamma(\tau))\,d\tau\right)$ .