On integrability of the Killing equation

Abstract

Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Since many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate equations of motion. Thus it has been desirable to formulate the integrability conditions of the Killing equation, which serve to determine the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. In this paper, we show the prologation for the Killing equation in a manner that uses Young symmetrizers. Using the prolonged equations, we provide the integrability conditions explicitly.

Distinguished curves and first integrals on Poincaré-Einstein and other conformally singular geometries

“…Geodesics and other distinguished curves play a basic and essential role in differential geometry and its applications [1,14,21,24,31] The determination and study of these can be enormously simplified if one has a available curve first integrals [1,6,14,25,29,8]. Certain conformally singular geometries such as Poincaré-Einstein manifolds have proved to have a central place in mathematical physics, geometric scattering, the AdS/CFT correspondence of physics, as well as in conformal geometry itself [10,22,32,11,12,20,27].…”

A special class of symmetric Killing 2-tensors

“…Even though already for valence two these constructions are most efficently formulated by means of Young symmetrizers, explicit descriptions avoiding this formalism are possible, see for example [11] (for valence two) and [24,25] (for arbitrary valence). For symmetric tensors of arbitrary valence, Y. Houri and others [14] have given an explicit construction of the Killing prolongation via a recursive construction using the technique of Young symmetrizers. Although their arguments can not immediately be generalised to other symmetry types, nevertheless one would conjecture that similar explicit constructions are possible for all symmetry types of tensors.…”

“…The prolongation of the Killing equation for abitrary symmetric tensors using the adjoint Young symmetrizers and projectors is given in [14]. However in comparison with (60)-(62) that approach has the disadvantage of the appearance of an additional Young symmetrizer (see [14, (7)-(9)]) complicating the formulas unnecessarily.…”

A criterion for the existence of Killing vectors in 3D

“…. , ∇ a W (6) T , (83) with 4 × 4 minors giving obstructions, playing a role similar to (7). Here W (i) are principal traces of the i-th powers of the Weyl tensor, considered as an endomorphism of Λ 2 T M. A criterion on existence of KVs in 4D should be based on this obstruction matrix.…”

“…While a principal approach was sketched in [4], it was only relatively recently that the final solution was found in [5], including specification of the number of KTs of order 2 depending on the curvature invariants of g ab . Criteria for the existence of higher order KTs in dimension 2 and KTs of order 2 in general dimension are overly complicated; see [6] for further discussion. In this paper, we devise an algorithm for computing the number of local KVs for Riemannian manifolds (M, g ab ) of dimension 3.…”