= Parallel vector field along a curve
{title2=$\nabla_{\dot\gamma}V=0$}
= Parallel vector fields along a curve
{synonym}
A <vector field> along a smooth curve is parallel if its <covariant derivative> along that curve is zero. Its value is the <parallel transport> of its initial value. A <Levi-Civita connection> preserves inner products under this transport, so the length of a parallel field is constant. Along a unit-speed curvature-one <geodesic>, the normal <Jacobi field> with zero initial value and initial derivative $a$ is $\sin t$ times the parallel transport of $a$.
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