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Flat pencils of metrics and integrable reductions of Lamé’s equations. (English. Russian original) Zbl 1006.53017

Russ. Math. Surv. 56, No. 2, 416-418 (2001); translation from Usp. Mat. Nauk 56, No. 2, 221-222 (2001).
The author proposes a method of solving a particular system of PDE’s which is obtained by reduction of so called Lamé equations related to the problem of compatibility of flat pseudo-Riemannian metrics. (Two such metrics \(g_1\) and \(g_2\) are said to be compatible iff the Levi-Civita connections and curvature tensors of any linear combination \(a_1g_1 + a_2g_2\) with constant coefficients \(a_i\) are of the form \(a_1\nabla_1 + a_2\nabla_2\) and \(a_1R_1 + a_2R_2\), \(\nabla_i\) and \(R_i\) being the Levi-Civita connections and curvature tensors of \(g_i\), \(i = 1, 2\)).

MSC:

53B21 Methods of local Riemannian geometry
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