Killing and conformal Killing tensors

Abstract

This thesis describes the prolongation connection of Killing tensors in terms of Young symmetrizers. The goal is to give an interpretation to sections of the prolongation bundle for Killing tensors on a manifold as generalized curvature tensors on the cone over that manifold. As a result, this method allows to treat the components of the prolongation bundle as a single object with well-understood symmetries. The developed formalism is then explored in three applications. The first result gives an isomorphism between the symmetric algebra of Killing tensors on a manifold of constant curvature and an algebra generated by parallel two-forms on the cone. That provides a geometric proof of the decomposition of Killing tensors on constant curvature manifolds and the Delong-Takeuchi-Thompson formula, previously obtained by Takeuchi and Thompson. Secondly, this technique, together with some branching rules for holonomy subgroups, yields a new characterization of Sasakian and 3-Sasakian manifolds in terms of Killing tensors satisfying additional curvature conditions. The third application is a new short proof of the result by Dairbekov and Sharafutdinov that the codimension of the zero set of a non-trivial, trace free, conformal Killing tensor is at least two. Throughout this work, special emphasis is placed on the representation theory of the appearing tensor bundles. Therefore, the Killing- and conformal Killing operators are introduced as Stein-Weiss operators. Since Young symmetrizers are a well-established tool in describing tensor representations this approach fits perfectly with the goals of the thesis. A natural consequence of this choice are new, geometric proofs of some established results. Besides those mentioned above these cover: (1) A Weitzenböck formula, which implies that all trace free, conformal Killing tensors on manifolds with non-positive sectional curvature are parallel. (2) The decomposition of occurring representations with respect to the reduced holonomy of a Riemannian product yields that the space of trace free, conformal Killing two-tensors on the product is generated by pullbacks of Killing one- and two-tensors on the factors. Furthermore, this thesis recasts the known examples of Killing tensors on compact Riemannian manifolds in the modern and coordinate free language of differential geometry. It is shown how the example found by Page and Pope generalizes to a construction on all Riemannian submersions with totally geodesic fibres. This technique provides non-parallel symmetric Killing two-tensors on compact Kähler manifolds. That contrasts the fact that on such n-dimensional manifolds do not exist non-parallel Killing forms of degree other than one or n-1. Furthermore, this construction gives a method to compute some eigenvalues of the Lichnerowicz-Laplace operator acting on symmetric two-tensors.