Geometric Mechanics and Control Seminar

Generalized Haantjes geometry and integrable systems

Speaker:  Piergiulio Tempesta (ICMAT - UCM)
Date:  Friday, 09 April 2021 - 15:30
Place:  Online - (ID: 912 3395 2494; Access code: 002326)

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. They represent a natural generalization of the well-known class of Nijenhuis tensors, which has a fundamental role in many geometric contexts. We introduce the family of symplectic-Haantjes manifolds (or ωH manifolds), as a natural setting where the notion of integrability can be formulated. We peove that the existence of a suitable Haantjes algebra of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. The theory of classical Hamiltonian systems admitting separating variables can also be formulated in the context of the ωH-structures. A special class of coordinates, called Darboux-Haantjes coordinates, enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We also propose a new, infinite class of generalized bivariate tensor fields, whose first representative is the known Frölicher--Nijenhuis bracket. This new family of tensors possesses many interesting algebraic-geometric properties. In particular, the bivariate Haantjes tensor is relevant in the characterization of Haantjes moduli of operator fields. Our main result is that the vanishing of a generalized Haantjes torsion is a sufficient condition for the integrability of the eigen-distributions of an operator field on a differentiable manifold. This new condition, which does not require the explicit knowledge of the spectral properties of a given operator field, generalizes the celebrated Haantjes theorem, because it provides us with an integrability criterion for operator fields whose standard Haantjes torsion is non-vanishing. Finally, as a contribution to the Courant problem for PDEs, we show that if an operator field has a vanishing higher-level Haantjes torsion, then it can be written, in a suitable local chart, in a block-diagonal form. Work in collaboration with D. Reyes Nozaleda and G. Tondo.