On Haantjes tensors for second-order superintegrable systems

The vanishing of the Haantjes tensor is an important property that has been linked, for instance, to the existence of separation coordinates and the integrability of systems of hydrodynamic type. We discuss the vanishing of the Haantjes tensor for operator fields that admit a large number of so-called conservation laws. In particular, we investigate Haantjes-zero Killing tensor fields that are associated with second-order superintegrable systems.

💡 Research Summary

The paper investigates the relationship between the vanishing of the Haantjes tensor and the presence of conservation laws in the context of second‑order superintegrable systems. Starting from a (pseudo‑)Riemannian manifold ((M,g)), the authors introduce a (1,1) operator field (A) and recall the definitions of its Nijenhuis tensor (N) and Haantjes tensor (H). They emphasize that the vanishing of (H) is linked to the integrability of the eigen‑distributions of (A), to the existence of separation coordinates, and to the integrability of certain hydrodynamic‑type PDEs.

The first central question (Question 1) asks whether the existence of (n+1) linearly independent conservation laws for an operator field on an (n)‑dimensional manifold forces (H) to be zero. By assuming a set of conservation laws generated by the quadratic function (u^{(0)}=\sum_{k=1}^{n}x_k^2) and the linear functions (u^{(m)}=x_m) ((m=1,\dots,n)), they show that the condition (d(A^\ast du^{(m)})=0) forces the operator to be a Hessian, (A^{ij}=\partial_i\partial_j f), for some scalar function (f). Two explicit examples in three dimensions illustrate the answer: with (f=x_1^3) the Haantjes tensor vanishes, while with (f=x_1^3+x_1x_2x_3) it does not. Hence the answer to Question 1 is negative; the mere presence of (n+1) conservation laws does not guarantee a Haantjes‑zero operator.

The authors then restrict attention to Killing tensor fields, i.e. symmetric ((0,2)) tensors (K) satisfying (\nabla_{(i}K_{jk)}=0). Via the metric, such tensors correspond to (1,1) operators, and the second question (Question 2) asks whether a Killing tensor admitting (n+1) independent conservation laws must have a vanishing Haantjes tensor. In three‑dimensional Euclidean space they consider the same four conservation laws as before and parametrize the full linear space of Killing tensors that satisfy them as a six‑parameter family \