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Einstein metrics with two-dimensional Killing leaves and their applications in physics. (English) Zbl 1382.53010

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 12th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 4–9, 2010. Sofia: Bulgarian Academy of Sciences. Geometry, Integrability and Quantization, 329-341 (2011).
Summary: Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian four-metrics admitting a non-abelian two dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a three dimensional Lie algebra of Killing fields with two dimensional leaves. Metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves.
For the entire collection see [Zbl 1245.00049].

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53Z05 Applications of differential geometry to physics
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