Compatible and almost compatible pseudo-Riemannian metrics

In this paper, notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics (this notion plays an important role in the theory of integrable systems of hydrodynamic type and the Dubrovin theory of Frobenius manifolds [1], see also [2]–[8]), are introduced and studied. Compatible metrics generate compatible Poisson structures of hydrodynamic type (these structures are local for flat metrics [9] and they are nonlocal if the metrics are not flat [10]–[14]) and their investigation is necessary for the theory of integrable systems of hydrodynamic type. In “nonsingular” case, when eigenvalues of pair of metrics are distinct, in this paper the complete explicit description of compatible and almost compatible metrics is obtained. The “singular” case of coinciding eigenvalues of pair of metrics is considerably more complicated for the complete analysis and has still not been completely studied. Nevertheless, the problem on two-component compatible flat metrics is completely investigated. All such pairs, both “nonsingular” and “singular”, are classified by ours. In this paper we present the complete description of nonsingular pairs of two-component flat metrics. The problems of classification of compatible flat metrics and compatible metrics of constant Riemannian curvature are of particular interest, in particular, from the viewpoint of the theory of Hamiltonian systems of hydrodynamic type. More detailed classification results for these problems will be published in another paper. In the given paper we prove that the approach proposed by Ferapontov in [4] for the study of flat pencils of metrics can be also applied (with the corresponding modifications and corrections) to pencils of metrics of constant Riemannian curvature and to the general compatible Riemannian and pseudo-Riemannian metrics. We also correct a mistake which is in [4] in the criterion of compatibility of local nondegenerate Poisson structures of hydrodynamic type (or, in other words, compatibility of flat metrics). We shall use both contravariant metrics g(u) with upper indices, where u = (u, ..., u) are local coordinates, 1 ≤ i, j ≤ N , and covariant metrics gij(u) with lower indices, g (u)gsj(u) = δ j . The indices of coefficients of the Levi–Civita connections Γ i jk(u) (the Riemannian connections generated by the corresponding metrics) and tensors of Riemannian curvature R jkl(u) are raised and lowered by the metrics corresponding to them: