Geodesic vector field

In a pseudo-Riemannian manifold $(M,g)$, a vector field $X$ is a geodesic vector field if

being $\mathrm{\nabla }$ the Levi-Civita connection.

They are also known as autoparallel vector fields (see @Carinena2023) and also unit geodesic vector fields (in this case, geodesic vector field refers to pregeodesic vector fields).

It can be shown that this definition is equivalent to require that the integral curves of $X$ be geodesic (see @Carinena2023 page 10).