Geodesic
Smooth kinematic-type manifolds
Teoretičeskaâ i matematičeskaâ fizika, Tome 119 (1999) no. 2, pp. 264-281

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We propose a general approach for describing different causality-type relations on smooth manifolds. The causality structure can be defined either axiomatically (by a cone in the tangent space) or by a pseudometric with the signature $(+-\cdots-)$ or $(+-\cdots-0\cdots0)$. In the latter case, the manifold acquires the structure of a fibered space with “absolute simultaneity” fibers. The smooth structure (atlas) of the manifold is directly related to its causal structure. 
@article{TMF_1999_119_2_a2,
     author = {V. R. Krym},
     title = {Smooth kinematic-type manifolds},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {264--281},
     publisher = {mathdoc},
     volume = {119},
     number = {2},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1999_119_2_a2/}
}
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V. R. Krym. Smooth kinematic-type manifolds. Teoretičeskaâ i matematičeskaâ fizika, Tome 119 (1999) no. 2, pp. 264-281. http://geodesic.mathdoc.fr/item/TMF_1999_119_2_a2/