Beyond recursion operators. (English) Zbl 1426.53001
Kielanowski, Piotr (ed.) et al., Geometric methods in physics XXXVI. Workshop and summer school, Białowieża, Poland, July 2–8, 2017. Selected papers of the 36th workshop (WGMPXXXVI) and extended abstracts of lectures given at the 6th “School of geometry and physics”. Cham: Birkhäuser. Trends Math., 167-180 (2019).
Summary: We briefly recall the history of the Nijenhuis torsion of \((1, 1)\)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-Hamiltonian and symplectic Nijenhuis formalisms, the sequence of powers of the recursion operator being replaced by a family of commuting Haantjes operators.
For the entire collection see [Zbl 1417.53002].
For the entire collection see [Zbl 1417.53002].
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53D99 | Symplectic geometry, contact geometry |
| 53A55 | Differential invariants (local theory), geometric objects |
| 01A60 | History of mathematics in the 20th century |
| 17B70 | Graded Lie (super)algebras |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
