# On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type

## TL;DR

This paper proves that non-singular non-local bi-Hamiltonian systems of hydrodynamic type are diagonalizable and can be expressed in local coordinates where all related structures are diagonal, extending the theory of compatible metrics.

## Contribution

It establishes the diagonalizability of non-singular non-local bi-Hamiltonian hydrodynamic systems and existence of local coordinates simplifying their structures.

## Key findings

- Systems are diagonalizable in suitable local coordinates.
- All related matrix and metric objects are diagonal in these coordinates.
- The results extend the theory of compatible metrics and bi-Hamiltonian structures.

## Abstract

We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all the related matrix differential-geometric objects, namely, the matrix V^i_j(u) of this system of hydrodynamic type, the metrics g^{ij}_1(u) and g^{ij}_2(u) and the affinors (w_{1, n})^i_j(u) and (w_{2,n})^i_j(u) of the non-singular non-local bi-Hamiltonian structure of this system, are diagonal in these local coordinates. The proof is a natural consequence of the general results of the theory of compatible metrics and the theory of non-local bi-Hamiltonian structures developed earlier by the present author.