Generalized Nijenhuis geometry and applications to Hamiltonian integrable systems

Abstract

We propose a new, infinite family of tensor fields, whose first representatives are the classical Nijenhuis and Haantjes tensors. We prove that the vanishing of a suitable higher-level 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 the considered operator, generalizes the celebrated Haantjes theorem, because it provides us with an effective integrability criterion applicable to the generic case of non-Nijenhuis and non-Haantjes tensors. We also propose a tensorial approach to the theory of classical Hamiltonian integrable systems, based on the geometry of Haantjes tensors. We introduce the family of symplectic-Haantjes manifolds as a natural setting where the notion of integrability can be formulated. In particular, the theory of separation of variables for classical Hamiltonian systems can also be formulated in the context of our new geometric structures. References 1. P. Tempesta, G. Tondo, Contemporary Mathematics, AMS (2023) (to appear) 2. D. Reyes, P. Tempesta, G. Tondo, J. Nonlinear Science 33, 35 (2023) 3. P. Tempesta, G. Tondo, Communications in Mathematical Physics 389, 1647-1671 (2022) 4. P. Tempesta, G. Tondo, Annali Mat. Pura Appl. 201, 57-90 (2022) 5. P. Tempesta, G. Tondo, J. Geometry and Physics 160, 103968 (2021)