Geodesic
The Einstein equations in the absence of matter fields on a 5-manifold with the causal structure
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 155-166

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We consider a 5-manifold with restricted smooth structure and appropriate group of coordinate transfirmations which includes general relativity and gauge transformations. An explicit expression for the Riemannian curvature of a 4-dimensional vector distribution leads to classical Einstein and Maxwell equations. 
@article{ZNSL_1999_261_a10,
     author = {V. R. Krym},
     title = {The {Einstein} equations in the absence of matter fields on a 5-manifold with the causal structure},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {155--166},
     publisher = {mathdoc},
     volume = {261},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a10/}
}
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V. R. Krym. The Einstein equations in the absence of matter fields on a 5-manifold with the causal structure. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 155-166. http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a10/