Parallel vector field
= Parallel vector field
{title2=$\nabla_YX=0\text{ for every }Y$}
A <vector field> is parallel for a specified <connection on a vector bundle> on the <tangent bundle> when its <covariant derivative> in every direction vanishes. Its values along every <smooth curve> are related by <parallel transport>. For a <Levi-Civita connection>, its length is constant on each <connected component>, since the connection preserves the <Riemannian metric>.