Following the example of the Lie derivative, we can consider parallel transport of a vector \({v}\) in the direction \({v}\) as generating a local flow. More precisely, for any vector \({v}\) at a point \({p\in M}\), there is a curve \({\phi_{v}(t)}\), unique for some \({-\varepsilon<t<\varepsilon}\), such that \({\phi_{v}(0)=p}\) and \({\dot{\phi}_{v}\left(t\right)=\parallel_{\phi}(v)}\), the last expression indicating that the tangent to \({\phi_{v}}\) at \({t}\) is equal to the parallel transport of \({v}\) along \({\phi_{v}}\) from \({\phi_{v}(0)}\) to \({\phi_{v}(t)}\). This curve is called an autoparallel geodesic (AKA geodesic, autoparallel curve), and its tangent vectors are all parallel transports of each other. This means that for all tangent vectors \({v}\) to the curve, \({\nabla_{v}v=0}\), so that autoparallel geodesics are “the closest thing to straight lines” on a manifold with parallel transport.

Expressing an autoparallel geodesic as a parametrized curve \({C^{\mu}(t)}\) with tangent \({v^{\mu}\left(t\right)\equiv\dot{C}^{\mu}\left(t\right)}\) in given coordinates, we can write

\begin{aligned}\nabla_{v}v & =v^{\lambda}\left(\partial_{\lambda}v^{\mu}+\Gamma^{\mu}{}_{\sigma\lambda}v^{\sigma}\right)\\
& =\partial_{v}\left(v^{\mu}\right)+\Gamma^{\mu}{}_{\sigma\lambda}v^{\sigma}v^{\lambda}\\
& =\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{d}C^{\mu}}{\mathrm{d}t}\right)+\Gamma^{\mu}{}_{\sigma\lambda}\frac{\mathrm{d}C^{\sigma}}{\mathrm{d}t}\frac{\mathrm{d}C^{\lambda}}{\mathrm{d}t}\\
& =\frac{\mathrm{d}^{2}C^{\mu}}{\mathrm{d}t^{2}}+\Gamma^{\mu}{}_{\sigma\lambda}\frac{\mathrm{d}C^{\sigma}}{\mathrm{d}t}\frac{\mathrm{d}C^{\lambda}}{\mathrm{d}t}=0,
\end{aligned}

where the last line is called the geodesic equation, and in the third line we use the fact that the change of the 0-form \({v^{\mu}}\) in the \({v}\) direction is equal to the derivative of the function \({v^{\mu}\left(t\right)}\) with respect to \({t}\).

Now following the example of Lie groups, we can define the exponential map at \({p}\) to be \({\mathrm{exp}(v)\equiv\phi_{v}(1)}\), which will be well-defined for values of \({v}\) around the origin that map to some \({U\subset M}\) containing \({p}\). Finally, choosing a basis for \({T_{p}U}\) provides an isomorphism \({T_{p}U\cong\mathbb{R}^{n}}\), allowing us to define geodesic normal coordinates (AKA normal coordinates) \({\mathrm{exp}^{-1}\colon U\rightarrow\mathbb{R}^{n}}\). It can be shown (see [13] Vol. 1 pp148-149) that in a coordinate frame at the origin \({p}\) of geodesic normal coordinates, we have \({\Gamma^{\lambda}{}_{\mu\sigma}=-\Gamma^{\lambda}{}_{\sigma\mu}}\); this implies that for zero torsion (to be defined here), the connection coefficients vanish at \({p}\).

The above depicts how geodesic normal coordinates at \({p}\) map points on a manifold to vectors at \({p}\) tangent to the geodesic passing through both points. In the figure \({\mathrm{exp}(2v)=\phi_{v}(2)}\), so the coordinate of the point \({\phi_{v}(2)\in M}\) is \({2v\in T_{p}M}\).

This is the third time we have utilized the concept of the flows (AKA integral curves, field lines, streamlines, trajectories, orbits) of vector fields. The relationships between these three situations are summarized below.