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

Abstract

We prove a conjecture formulated by Bolsinov, Konyaev and Matveev in [7] stating that, integrability of a system of hydrodynamic type ${\mathbf{u}}_t=A(\mathbf{u}){\mathbf{u}}_x$ with $\mathfrak{gl}$-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 $\mathfrak{gl}$-regular operator field can be written as ${\mathbf{u}}_t=X(\mathbf{u})\circ {\mathbf{u}}_x$ where $X$ is a vector field and $\circ$ is a commutative associative product satisfying Hertling-Manin conditions.

19 pages