The Variety of Integrable Killing Tensors on the 3-Sphere

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 $K_4$.

References in corpus (5)

• Superintegrable Systems in Darboux spaces
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
• Superintegrability in a non-conformally-flat space
• Separation coordinates, moduli spaces and Stasheff polytopes

Cited by in corpus (9)

• Killing and Conformal Killing tensors
• Invariant classification of second-order conformally flat superintegrable systems
• Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Orthogonal separation of variables for spaces of constant curvature
• Are Orthogonal Separable Coordinates Really Classified?
• Separation coordinates, moduli spaces and Stasheff polytopes
• Curvature and the c-projective mobility of Kaehler metrics with hamiltonian 2-forms
• Nijenhuis Integrability for Killing Tensors