Geodesic
Einstein's equations on a~$4$-manifold of conformal torsion-free connection
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 3, pp. 393-408

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The main defect of Einstein equations – non geometrical right part – is eliminated. The key concept of equidual tensor is introduced. It appeared to be in a close relation both with Einstein's equations, and with Yang–Mills equations. The criterion of equidual basic affinor of conformal connection manifold without torsion is received. Decomposition of the basic affinor into a sum of equidual, conformally invariant and irreducible summands is found. A. Z. Petrov's algebraic classification is generalized. Einstein equations are given a new variational foundation and their geometrical nature is found. Geometric sense of acceleration and dilatation gauge transformations is specified.
Keywords: Einstein equations, Yang–Mills equations, Hodge operator, Maxwell's equations, manifold of conformal connection with torsion and without torsion. 
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     author = {Leonid N. Krivonosov and Vyacheslav A. Luk'yanov},
     title = {Einstein's equations on a~$4$-manifold of conformal torsion-free connection},
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     url = {http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a11/}
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Leonid N. Krivonosov; Vyacheslav A. Luk'yanov. Einstein's equations on a~$4$-manifold of conformal torsion-free connection. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 3, pp. 393-408. http://geodesic.mathdoc.fr/item/JSFU_2012_5_3_a11/