= Geodesic
{title2=$\nabla_{\dot\gamma}\dot\gamma=0$}
{wiki}
A geodesic is a <curve> whose <tangent vector> is parallel along itself under the <Levi-Civita connection>. With an <affine parameter> it has zero covariant acceleration:
$$
\nabla_{\dot\gamma}\dot\gamma=0.
$$
This is the <Riemannian manifold> version of moving straight ahead with constant speed. Its coordinate form is the <geodesic equation>. In <Euclidean space> it becomes $\ddot\gamma=0$, giving straight lines. On an embedded <surface> with its <induced metric>, the ordinary acceleration of such a <curve> is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.
On a unit round <sphere>, the <great circle> $\gamma(t)=(\cos t,\sin t,0)$ has $\ddot\gamma(t)=-\gamma(t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi/3$ can be joined along that circle by the short arc of <length> $\pi/3$ or the long arc of <length> $5\pi/3$. Each admits a constant-speed <geodesic> parametrization, yet only the short arc realizes their <great-circle distance>.
\Image[/sphere-geodesic-short-and-long.png]
{title=Two great-circle arcs are geodesics, while only the shorter arc minimizes distance}
Every <geodesic> minimizes <arc length> on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the <cut locus>. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a <circular cylinder>, different winding gives the same distinction, made explicit by <Riemannian distance on a circular cylinder>.
This provides a useful division of work. The <geodesic equation> predicts a path from an initial point and <tangent vector>; comparing <arc lengths> decides which candidate solves a global shortest-route problem. Variations of <energy of a curve> give the equation as an <Euler-Lagrange equation>, offering a proof outline and a connection with <classical mechanics>. https://www.math.ucla.edu/~petersen/RG.pdf[Peter Petersen, Riemannian Geometry, chapter 5: geodesics, distance and first variation]{ref}.
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