Given a conformally nonflat Einstein space–time we define a fibration over it. The fibers of this fibration are elliptic curves (two-dimensional tori) or their degenerate counterparts. Their topology depends on the algebraic type of the Weyl tensor of the Einstein metric. The fibration is a double branched cover of the bundle 𝒫 of null direction over the space–time and is equipped with six linearly independent one-forms which satisfy a certain relatively simple system of equations.
Topics
Riemannian geometry
REFERENCES
1.
P.
Nurowski
, “On a certain formulation of the Einstein equations
,” J. Math. Phys.
39
, 5477
(1998
).2.
3.
R. Penrose (private communications).
4.
E.
Cartan
, “Sur les espaces conformes generalises at l’universe optique
,” Compt. Rendus Acad. Sci. Paris
174
, 857
–859
(1922
).5.
R.
Penrose
, “A spinor approach to general relativity
,” Ann. Phys. (N.Y.)
10
, 171
–201
(1960
).6.
A. Z.
Petrov
, “Classification of spaces defining gravitational fields
,” Sci. Not. Kazan State Univ.
114
, 55
–69
(1954
).
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© 1998 American Institute of Physics.
1998
American Institute of Physics
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