Geodesic
Levi-Chivita connections on degenerate pseudo-Riemannian manifolds
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 134-141
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Smooth manifolds with bilinear symmetric degenerate form with constant signature in every tangent space are studied. Such manifolds are used in physics for constructing theories of united interactions including gravitation. Necessary and sufficient conditions for existence of a Levi-Chivita connection on such a manifold are established. 
@article{ZNSL_1998_252_a11,
     author = {S. E. Kozlov},
     title = {Levi-Chivita connections on degenerate {pseudo-Riemannian} manifolds},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {134--141},
     year = {1998},
     volume = {252},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a11/}
}
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S. E. Kozlov. Levi-Chivita connections on degenerate pseudo-Riemannian manifolds. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 134-141. http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a11/