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Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations, and integrability. (English. Russian original) Zbl 1083.53041

Funct. Anal. Appl. 36, No. 3, 196-204 (2002); translation from Funkts. Anal. Prilozh. 36, No. 3, 36-47 (2002).
Let \(g'\) and \(g''\) be two pseudo-Riemannian metrics. The metrics \(g'\) and \(g''\) are said to be compatibles if for any linear combination \(g = ag' + bg''\), where \(a, b\) are arbitrary constants such that the \(\det(g)\) is non-zero, the coefficients of the Levi-Civita connections and the components of the corresponding Riemannian curvature tensors are related by the same linear formula; in this case, the metrics \(g'\) and \(g''\) form a pencil of compatible metrics.
In a previous paper [O. I. Mokhov, Funct. Anal. Appl. 35, No. 2, 100–110 (2001); translation from Funkts. Anal. Prilozh. 35, No. 2, 24–36 (2001; Zbl 1005.53016); see also math.DG/0005051], the author considered the description problem for flat pencils of metrics; also, a method for integrating the nonlinear equations for nonsingular (i.e. the eigenvalues of the pair \(g', g''\) are distinct) flat pencils of metrics was proposed. In the present paper, the description problem for nonsingular pencils of constant curvature is solved. The integrability of the nonlinear equations by the inverse scattering method is proved. In terms of Ferapontov’s method [E. V. Ferapontov, J. Phys. A, Math. Gen. 34, No. 11, 2377–2388 (2001; Zbl 1010.37044), math.DG/0005221], the Lax Pair with a spectral parameter is derived from the linear problem for the system of nonlinear equations describing all orthogonal curvilinear coordinate systems in an \(n\)-dimensional space of constant curvature.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
53B20 Local Riemannian geometry
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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