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Existence of Riemannian invariants for integrable systems of hydrodynamic type

Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev

TL;DR

The work addresses when a hyperbolic system of hydrodynamic type with $n$ mutually symmetric operator fields $K_1,\dots,K_n$ admits $n$ Riemann invariants. It proves that, assuming a linear combination $\sum c_i K_i$ has $n$ distinct real eigenvalues, there exists a local coordinate system in which all $K_i$ are diagonal, linking symmetry richness to integrability. The proof constructs joint eigenvector fields $v_i$ and shows $[v_i,v_j]$ lies in their span via algebraic brackets and a pointwise normal form, thereby obtaining diagonalization. The discussion extends to complex spectra and Jordan blocks, with computer-algebra checks suggesting a broader Haantjes-torsion vanish condition and a conjecture connecting $\mathrm{gl}$-regular combinations to vanishing Haantjes torsion, reinforcing the Tsarev-integrable perspective.

Abstract

We show that for a hyperbolic system of hydrodynamic type admitting n symmetries, there exists a coordinate system in which the generator of the system and all the symmetries are diagonal.

Existence of Riemannian invariants for integrable systems of hydrodynamic type

TL;DR

The work addresses when a hyperbolic system of hydrodynamic type with mutually symmetric operator fields admits Riemann invariants. It proves that, assuming a linear combination has distinct real eigenvalues, there exists a local coordinate system in which all are diagonal, linking symmetry richness to integrability. The proof constructs joint eigenvector fields and shows lies in their span via algebraic brackets and a pointwise normal form, thereby obtaining diagonalization. The discussion extends to complex spectra and Jordan blocks, with computer-algebra checks suggesting a broader Haantjes-torsion vanish condition and a conjecture connecting -regular combinations to vanishing Haantjes torsion, reinforcing the Tsarev-integrable perspective.

Abstract

We show that for a hyperbolic system of hydrodynamic type admitting n symmetries, there exists a coordinate system in which the generator of the system and all the symmetries are diagonal.
Paper Structure (3 sections, 2 theorems, 21 equations)

This paper contains 3 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:1']}
  3. Discussion of the complex eigenvalues and of Jordan blocks

Key Result

Theorem 1

Let $K_1, \dots, K_n$ be $n$-dimensional mutual symmetries. Assume that at a point $p$, they are linearly independent and there exists a linear combination $\sum c_iK_i$ having $n$ distinct real eigenvalues. Then, in a neighborhood of $p$ there exists a local coordinate system such that all $K_i$ ar

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2
  • proof
  • Conjecture 1