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Invariant prolongation of the Killing tensor equation

  • A. Rod Gover
  • Thomas Leistner
Article

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.

Keywords

Integrability Hidden symmetries Projective differential geometry Riemannian manifolds Affine manifolds 

Mathematics Subject Classification

Primary 53B10 Secondary 53A20 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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