The Variety of Integrable Killing Tensors on the 3-Sphere

Konrad Schöbel

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton–Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere $S^3$ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on $S^3$ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron $K_4$.

Détail

Keywords: separation of variables; Killing tensors; Stäckel systems; integrability; algebraic curvature tensors.

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