Abstract
We consider the pseudo-Euclidean space (R n, g), with n ≥ 3 and g ij = δ ij ε i , ε i = ±1, where at least one ε i = 1 and nondiagonal tensors of the form T = Σ ij f ij dx i dx j such that, for i ≠ j, f ij (x i , x j ) depends on x i and x j . We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R n, on the n-dimensional torus T n and on cylinders T k×R n-k, that solve the Ricci equation or the Einstein equation.
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References
J. Cao and D. DeTurck, The Ricci curvature equation with rotational symmetry, American Journal of Mathematics 116 (1994), 219–241.
D. DeTurck, Existence of metrics with prescribed Ricci Curvature: Local Theory, Inventiones Mathematicae 65 (1981), 179–207.
D. DeTurck, Metrics with prescribed Ricci curvature, Seminar on Differential Geometry, in Annals of Mathematics Studies, Vol. 102, Princeton University Press, Princeton, NJ, 1982, pp. 525–537.
D. DeTurck, The Cauchy problem for Lorentz metrics with Prescribed Ricci curvature, Compositio Mathematica 48 (1983), 327–349.
D. DeTurck and H. Goldschmidt, Metrics with Prescribed Ricci Curvature of Constant Rank, Advances in Mathematics 145 (1999), 1–97.
D. DeTurck and W. Koiso, Uniqueness and non-existence of metrics with prescribed Ricci curvature, Annales de l’Institut Henry Poincaré. Analyse Non Linéaire 1 (1984), 351–359.
R. S. Hamilton, The Ricci curvature equation, in Seminar on nonlinear partial differential equations (Berkeley, California, 1983), Mathematical Science Research Institute Publications, Springer, New York, 1984, pp. 47–72.
J. Lohkamp, Metrics of negative Ricci curvature, Annals of Mathematics 140 (1994), 655–683.
R. Pina, Conformal Metrics and Ricci Tensors in the Hyperbolic space, Matemática Contemporâ nea 17 (1999), 254–262.
R. Pina and K. Tenenblat, Conformal Metrics and Ricci Tensors in the pseudo-Euclidean space, Proceedings of the American Mathematical Society 129 (2001), 1149–1160.
R. Pina and K. Tenenblat, On metrics satisfying equation R ij -Kg ij /2 = T ij for constant tensors T, Journal of Geometry and Physics 40 (2002), 379–383.
R. Pina and K. Tenenblat, Conformal Metrics and Ricci Tensors on the Sphere Proceedings of the American Mathematical Society 132 (2004), 3715–3724.
R. Pina and K. Tenenblat, On the Ricci and Einstein equations on the pseudo-Euclidean and hyperbolic spaces, Differential Geometry and its Applications 24 (2006), 101–107.
R. Pina and K. Tenenblat, A class of solutions of the Ricci and Einstein equations, Journal of Geometry and Physics 57 (2007), 881–888.
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein field equations, Cambridge University Press, Cambridge, MA, 2003.
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Pina, R., Tenenblat, K. On solutions of the Ricci curvature equation and the Einstein equation. Isr. J. Math. 171, 61–76 (2009). https://doi.org/10.1007/s11856-009-0040-y
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DOI: https://doi.org/10.1007/s11856-009-0040-y