Parallel vector field along a curve (${\nabla }_{\dot{\gamma }}V=0$)

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 $a$.