On compatible diagonal metrics

Abstract

In this note the well-known important problem of a complete description of compatible diagonal metrics is solved. In 2000 (see [1] and [2]) Mokhov obtained a complete explicit description of pairs of compatible metrics for which all eigenvalues are distinct. In the case of distinct eigenvalues such a pair of metrics can be simultaneously diagonalized and, as shown in [1] and [2], it is compatible if and only if the Nijenhuis tensor of the affinor associated with this pair of metrics vanishes. This made it possible in [1] and [2] to describe all such compatible metrics explicitly. The general case of pairs of compatible diagonal metrics that have coincident eigenvalues has remained unexplored despite its importance for applications. This case is completely investigated in this work. The general case of describing all pairs of compatible metrics remains an open problem to date. Compatible metrics play an important role in the theory of integrable systems, the Hamiltonian and bi-Hamiltonian theory of systems of hydrodynamic type, integrable hierarchies, the theory of Frobenius manifolds and their generalizations, the theory of multidimensional Poisson brackets, differential geometry and mathematical physics (see [3]–[12] and the review paper [13]). The general notion of compatible metrics was introduced by Mokhov in [1] and [2], and was motivated by the study of the compatibility conditions for local and non-local Poisson structures of hydrodynamic type, the theory of which was developed by Dubrovin and Novikov ([14], local theory) and Mokhov and Ferapontov ([15] and [16], non-local theory) for the purposes of the theory of systems of hydrodynamic type. Recall that a pair of contravariant (Riemannian or pseudo-Riemannian) metrics g 1 (u) and g ij 2 (u) is called almost compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Christoffel symbols corresponding to these metrics (the compatibility condition for the Levi-Civita connections of these metrics): Γ λ1,λ2;k(u) = λ1Γ ij 1;k(u) + λ2Γ ij 2;k(u), where Γ λ1,λ2;k(u) = g is λ1,λ2 (u)Γjλ1,λ2;sk(u), Γ ij 1;k(u) = g is 1 (u)Γ j 1;sk(u), and Γ ij 2;k(u) = g 2 (u)Γ j 2;sk(u). A pair of almost compatible metrics g ij 1 (u) and g ij 2 (u) is called compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Riemann curvature tensors corresponding to these metrics (the compatibility condition for the curvatures of these metrics):