Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of signature $(+,+,+,…,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(−,+,+,…,+).$

Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of signature $(+,+,+,…,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(−,+,+,…,+).$

Lorentzian geometry is analogous to Euclidean geometry (as Minkowski space is analogous to a Euclidean space), a Lorentzian manifold is analogous to a Riemannian manifold. Thus, one might use ‘Lorentzian geometry’ analogously to Riemannian geometry (and insist on Minkowski geometry for our topic here), but usually one skips all the way to pseudo-Riemannian geometry (which studies pseudo-Riemannian manifolds, including both Riemannian and Lorentzian manifolds).

See also: nLab article Lorentzian geometry, or Wikipedia: Pseudo-Riemannian manifold