The Variety of Integrable Killing Tensors on the 3Sphere
Abstract
Integrable Killing tensors are used to classify orthogonal coordinates in which the classical HamiltonJacobi 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 algebrogeometric 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.
 Publication:

SIGMA
 Pub Date:
 July 2014
 DOI:
 10.3842/SIGMA.2014.080
 arXiv:
 arXiv:1205.6227
 Bibcode:
 2014SIGMA..10..080S
 Keywords:

 separation of variables; Killing tensors; Stäckel systems; integrability; algebraic curvature tensors;
 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory;
 53A60 (Primary) 14H10;
 14M12 (Secondary)
 EPrint:
 SIGMA 10 (2014), 080, 48 pages