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On integrability of the Killing equation

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arxiv 1704.02074 v3 pith:H5YW53IP submitted 2017-04-07 gr-qc hep-thmath-phmath.DGmath.MP

classification gr-qchep-thmath-phmath.DGmath.MP
keywords killingequationequationsintegrabilityconditionsfieldsfirstgeodesic
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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. Many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, and thus solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate the equations of motion. In this paper we attempt to formulate the integrability conditions of the Killing equation, which serve to put an upper bound on the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. To this end, we first show the prologation for the Killing equation in a manner that uses Young symmetrizers. Then, using the prolonged equations, we provide the integrability conditions explicitly.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From (Hidden) Symmetries to Stealth Solutions

    gr-qc 2026-06 unverdicted novelty 6.0 of 10

    Killing-Yano tensors generate p-form stealth solutions in a bumblebee-type Proca theory with fine-tuned curvature terms on arbitrary backgrounds.

  2. On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime

    gr-qc 2019-08 accept novelty 5.0 of 10

    Using the Eisenhart-Duval lift, the authors reproduce previously known commuting symmetry operators for the Maxwell (LFKK) equation on Kerr-NUT-(A)dS spacetimes in covariant form, and construct the analogous operator ...

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