Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations and integrability

In the present paper, the nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. These results were announced in our previous paper [1]. For the flat pencils of metrics the corresponding statements and proofs were presented in the present author’s work [2], [3], where the method of integrating the nonlinear equations for the nonsingular flat pencils of metrics was proposed. In [4] the Lax pair for the nonsingular flat pencils of metrics was demonstrated. This Lax pair is generalized to the case of arbitrary nonsingular pencils of metrics of constant Riemannian curvature (see examples and interesting applications in [4]). In this paper, it is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvature are described by special integrable reductions of nonlinear equations defining orthogonal curvilinear coordinate systems in the spaces of constant curvature. Note that the problem of description for the pencils of metrics of constant Riemannian curvature is equivalent to the problem of description for compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets [5]) playing an important role in the theory of systems of hydrodynamic type. Recall that two pseudo-Riemannian contravariant metrics g 1 (u) and g ij 2 (u) are called compatible if for any linear combination of these metrics g(u) = λ1g ij 1 (u)+λ2g ij 2 (u), where λ1 and λ2 are arbitrary constants for which det(g (u)) 6≡ 0, the coefficients of the corresponding Levi-Civita connections and the components of the corresponding tensors of Riemannian curvature are related by the same linear formula: Γ k (u) = λ1Γ ij 1,k(u) + λ2Γ ij 2,k(u) and R kl(u) = λ1R ij 1,kl(u)+λ2R ij 2,kl(u) (in this case, we shall say also that the metrics g ij 1 (u) and g 2 (u) form a pencil of metrics) [1]. Flat pencils of metrics, that is nothing but compatible nondegenerate local Poisson brackets of hydrodynamic type (compatible Dubrovin– Novikov brackets [6]), were introduced in [7]. Two pseudo-Riemannian contravariant metrics g 1 (u) and g ij 2 (u) of constant Riemannian curvature K1 and K2 respectively are called compatible if any linear combination of these metrics g(u) = λ1g ij 1 (u)+λ2g ij 2 (u), where λ1 and λ2 are arbitrary constants for which det(g (u)) 6≡ 0, is a metric of constant Riemannian