Killing tensors in spaces of constant curvature

Abstract

A Killing tensor is one possible way of generalizing the notion of a Killing vector on a Riemannian or pseudo-Riemannian manifold. It is explained how Killing tensors may be identified with functions that are homogeneous polynomials in the fibers on the associated cotangent bundle. As such, Killing tensors may be identified with first integrals of the Hamiltonian geodesic flow, which are homogeneous polynomials in the momenta. Again using this identification, it is shown that in flat spaces the dimension of the vector space of Killing tensors is maximal and that the Killing tensors are generated by the Killing vectors. Finally, using Riemann’s model for the metric in spaces of constant curvature, a comparison argument is used to show that similar results are valid in that more general context.