Haantjes torsion and integrability: a proof of Bolsinov-Konyaev-Matveev's conjecture

2026/07/31 by Alessandro Arsie, Paolo Lorenzoni
Physics and Astronomy · Mathematics · #math-ph #math.DG #math.MP

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19 pages

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

We prove a conjecture formulated by Bolsinov, Konyaev and Matveev in [7] stating that, integrability of a system of hydrodynamic type \bf u_t=A(\bf u) \bf u_x with \mathfrakgl-regular A at a point p implies the vanishing of the Haantjes tensor of A and of all its symmetries in a neighborhood of p. As a consequence, leveraging on the result of [8], in a neighbourhood of an algebraically generic point, any integrable system of hydrodynamic type defined by a \mathfrakgl-regular operator field can be written as \bf u_t=X(\bf u)∘ \bf u_x where X is a vector field and ∘ is a commutative associative product satisfying Hertling-Manin conditions.

Citations

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