The Variety of Integrable Killing Tensors on the 3-Sphere

Abstract

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 .

Killing and conformal Killing tensors

“…In [26] the class of special conformal Killing tensors was introduced. These tensors were defined as symmetric 2-tensors K ∈ Γ(Sym 2 TM) satisfying the equation…”

“…Thus solutions of Equation (12) are automatically conformal Killing tensors. The tensor k also satisfies δK = −(n + 1) k and it is easily proved thatK := K − tr(K) g is a Killing tensor, which is called special Killing tensor in [26]. Moreover the map K →K is shown to be injective and equivariant with respect to the action of the isometry group.…”

“…in connection with geometric inverse problems, integrable systems and Einstein-Weyl geometry, cf. [5], [8], [9], [13], [15], [23], [26].…”

Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

“…In addition, using the fact that spaces considered admit isometry groups of maximal dimension, they develop a graphical calculus based on Lie group theory which summarizes the complete solution. Their calculus has recently been reinterpreted from an algebraic geometry point of view by Schöbel [38]. Waksjö and Rauch-Wojciechowski in [41] used this procedure to solve the last two fundamental Problems (2) and (3) for n-dimensional Euclidean and spherical spaces.…”

“…Indeed, one can show that the general Killing tensor, K, in a space of constant curvature is a sum of symmetrized products of the Killing vectors of the space [40]. An invariant condition that K have normal vector fields is that it satisfies the Tonolo-Schouten conditions or the equivalent Haantjes condition [13,29] both of which are non-linear in the coefficients of K. The general solution of these equations for S 3 is given in [38]. However, the solution for arbitrary n appears impossible.…”