Invariant prolongation of the Killing tensor equation

Abstract

The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolongation to a linear connection. This connection has the property that parallel sections are in 1-1 correspondence with solutions of the Killing equation. Moreover this connection is projectively invariant and is derived entirely using the projectively invariant tractor calculus which reveals also further invariant structures linked to the prolongation.

24 pages

References in corpus (5)

• Black holes, hidden symmetries, and complete integrability
• Quantum integrability of quadratic Killing tensors
• The conformal Killing equation on forms -- prolongations and applications
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• On integrability of the Killing equation

Cited by in corpus (4)

• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
• A special class of symmetric Killing 2-tensors
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries