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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 $(−,+,+,…,+).$
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Tensor Field Decomposition in Space time
See also
MR0317200 Barbance, Christiane Sur les tenseurs symétriques. (French) C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A387–A389.