Compatible flat metrics
by
O. I. Mokhov
Centre for Nonlinear Studies, Moscow, Russia

Oral Communication

We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general N-component case. We present the scheme of integrating the nonlinear equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on the reducing this problem to a special reduction of the Lame equations and the using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).

Definition 1. Two pseudo-Riemannian contravariant metrics g₁^ij (u) and g₂^ij (u) are called compatible if for any linear combination of these metrics

gij (u) = a1 g1ij (u) + a2 g2ij (u),

(1)

where a₁ and a₂ are arbitrary constants such that det( g^ij (u) ) =/= 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:

Gijk (u) = a1 G1, kij (u) +a2 G2, kij (u),

(2)

Rklij (u) = a1 R1, klij (u)+ a2 R2, klij (u).

(3)

We shall also say in this case that the metrics g₁^ij (u) and g₂^ij (u) form a pencil of metrics.

Definition 2. Two pseudo-Riemannian metrics g₁^ij (u) and g₂^ij (u) are called nonsingular pair of metrics if the eigenvalues of this pair of metrics, that is, the roots of the equation

det ( g1ij (u) - z g2ij (u)) = 0,

are distinct.

In the case if the metrics g₁^ij (u) and g₂^ij(u) are flat, that is, R_{1, jkl}ⁱ (u) = R_{2, jkl}ⁱ (u) = 0, relation (3) is equivalent to the condition that an arbitrary linear combination of the flat metrics g₁^ij (u) and g₂^ij(u) is also a flat metric and Definition 1 is equivalent to the well-known definition of a flat pencil of metrics or, in other words, a compatible pair of local nondegenerate Poisson structures of hydrodynamic type.

Theorem 1. 1) If for any linear combination (1) of two metrics g₁^ij (u) and g₂^ij (u) condition (2) is fulfilled, then the Nijenhuis tensor of the affinor

vij (u) = g1is (u) g2, sj (u),

that is,

Nkij (u) = vsi (u) (vkj)us- vsj (u) (vki)us +vks (u) (vsi)uj -vks (u) (vsj)ui,

vanishes.

2) If a pair of metrics g₁^ij (u) and g₂^ij (u) is nonsingular, then it follows from the vanishing of the Nijenhuis tensor of the affinor vⁱ_j (u) = g₁^is (u) g_{2, sj} (u) that the metrics g₁^ij (u) and g₂^ij (u) are compatible.

Theorem 2. An arbitrary nonsingular pair of metrics is compatible if and only if there exist local coordinates u = (u¹, ..., u^N) such that both the metrics are diagonal and, moreover, g^ij₂ (u) = gⁱ (u) \delta^ij and g^ij₁ (u) = (fⁱ (uⁱ))² gⁱ (u) \delta^ij, where fⁱ (uⁱ), i=1, ..., N, are arbitrary functions of single variables (of course, in the case of nonsingular pair of metrics, these functions are not equal to each other if they are constants and they are not equal identically to zero).

Let us consider the problem on nonsingular pairs of compatible flat metrics. It follows from Theorem 2 that it is sufficient to classify flat metrics of the form g₂^ij (u) = gⁱ (u) \delta^ij and g₁^ij (u) = (fⁱ (uⁱ))² gⁱ (u) \delta^ij, where fⁱ (uⁱ), i = 1, ..., N, are arbitrary functions of single variables.

The problem of description of diagonal flat metrics, that is, flat metrics g₂^ij (u) = gⁱ (u) \delta^ij, is a classical problem of differential geometry. This problem is equivalent to the problem of description of curvilinear orthogonal coordinate systems in a pseudo-Euclidean space and it was studied in detail and mainly solved in the beginning of the 20th century (Darboux). Locally, such coordinate systems are determined by n(n-1)/2 arbitrary functions of two variables (Bianchi, E. Cartan). Recently, Zakharov showed that the Lame equations describing curvilinear orthogonal coordinate systems can be integrated by the inverse scattering method.

Theorem 3. Nonsingular pairs of compatible flat metrics are described by the following integrable nonlinear equations which are the special reductions of the Lame equations:

(bij)uk = bik bkj,  i =/= j,  i =/= k,  j =/= k,

(4)

(bij)ui+(bji)uj+ å s =/= i, s =/= j  bsi bsj = 0,  i =/= j,

(5)

fi (ui) (fi (ui) bij)ui+fj (uj)(fj (uj)bji)uj+ å s =/= i, s =/= j  (fs)2 (us) bsi bsj = 0,  i =/= j,

(6)

where fⁱ (uⁱ), i=1, ..., N, are the given arbitrary functions of single variables.

The equations (4) and (5) are the famous Lame equations and the equation (6) defines a nontrivial nonlinear differential reduction of the Lame equations. Differential reductions of type (5) for equations (4) were studied by Zakharov and the Zakharov method can be applied successfully to our problem [1], [2].

[1] O.I.Mokhov. arXiv: math.DG/0005051. [2] O.I.Mokhov. arXiv: math.DG/0005081.

Paper reference: arXiv:math.DG/0005051, arXiv:math.DG/0005081

Date received: May 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-45.