It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes.
[1] It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since where
is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional manifold.
implies that the modulus of a vector defined on the tangent bundle to a certain point
of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.