Geodesics and normal coordinates

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 a geodesic, 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 geodesics are “the closest thing to straight lines” on a manifold with parallel transport.

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  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.

Lie derivativeLie groupCovariant derivative
Added structure on $${M}$$Vector field $${v}$$Group structure on pointsParallel transport $${\parallel_{C}(v)}$$
Vector field on $${M}$$$${v}$$Left-invariant vector fields $${A}$$Tangents to geodesics
Flow of vector fieldLocal flow $${v_{p}(t)}$$One-parameter subgroup $${\phi_{A}(t)}$$Geodesics $${\phi_{v}(t)}$$
Exponential map of flowLocal one-parameter diffeomorphism $${\Phi_{t}(v)\equiv v_{p}(t)}$$$${e^{A}\equiv\phi_{A}(1)}$$$${\mathrm{exp}(v)\equiv\phi_{v}(1)}$$
Diffeomorphism of $${\mathrm{exp}}$$Local in general, global if $${v}$$ is completeLocal to identityLocal to origin of $${\mathrm{exp}}$$
Vector transport by flowTangent map (only along $${v}$$)Tangent map along $${A}$$Parallel transport along geodesics $${\parallel_{\phi}(v)}$$
Vector derivative from transport$${L_{v}w}$$ is the difference between $${w}$$ and its transport by the local flow of $${v}$$$${[A,B]}$$ is the difference between $${B}$$ and its transport by the local flow of $${A}$$$${\nabla_{v}w}$$ is the difference between $${w}$$ and its parallel transport in the direction $${v}$$