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 map $s:I\to E$ such that $\pi\circ s=\gamma$. Using $\nabla$, one defines the covariant derivative of $s$ along $\gamma$ by

where $\tilde s$ is any smooth extension of $s$ to a neighborhood of $\gamma(t)$ in $M$. This is well-defined (independent of the choice of extension) by the locality and $C^\infty(M)$-linearity properties of a connection.

A section $s$ along $\gamma$ is called parallel along $\gamma$ if

Equivalently: 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

1. 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 $v(t)\equiv v_0$.

2. 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 $t$.

3. 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 $f(t)=f(t_0)\exp\!\left(-\int_{t_0}^t \alpha(\dot\gamma(\tau))\,d\tau\right)$.