Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 080, 48 pages      arXiv:1205.6227      https://doi.org/10.3842/SIGMA.2014.080

The Variety of Integrable Killing Tensors on the 3-Sphere

Konrad Schöbel
Institut für Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

Received November 14, 2013, in final form July 15, 2014; Published online July 29, 2014

Abstract
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 $K_4$.

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

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