Geodesic
Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 36-47

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The description problem is solved for compatible metrics of constant Riemannian curvature. Nonlinear equations describing all nonsingular pencils of compatible metrics of constant Riemannian curvature are derived and their integrability by the inverse scattering method is proved. In particular, a Lax pair with a spectral parameter is found for these nonlinear equations. We prove that all nonsingular pairs of compatible metrics of constant Riemannian curvature are described by special integrable reductions of the nonlinear equations defining orthogonal curvilinear coordinate systems in spaces of constant curvature.
Keywords: flat pencil of metrics, compatible metrics, metric of constant Riemannian curvature, nonlinear integrable equation
Mots-clés : Lax pair, compatible Poisson brackets. 
@article{FAA_2002_36_3_a3,
     author = {O. I. Mokhov},
     title = {Compatible {Metrics} of {Constant} {Riemannian} {Curvature:} {Local} {Geometry,} {Nonlinear} {Equations,} and {Integrability}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {36--47},
     publisher = {mathdoc},
     volume = {36},
     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2002_36_3_a3/}
}
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O. I. Mokhov. Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 36-47. http://geodesic.mathdoc.fr/item/FAA_2002_36_3_a3/