red bullet Killing Tensor Fields and Killing Forms  

In General > s.a. killing vector fields / conservation laws.
$ Killing tensor: An n-th rank Killing tensor is a symmetric covariant tensor, Kab... c = K(ab... c), such that ∇(m Kab... c) = 0.
$ Killing form: A differential form ωab... c = ω[ab... c] whose covariant derivative is totally skew-symmetric.
* Conserved quantities: While Killing vectors give the linear first integrals of the geodesic equations, Killing tensors give the quadratic, cubic, and higher-order first integrals; For every Killing tensor Kab... c, umm (Kab... c uaub ··· uc) = 0, if ua is tangent to affinely parametrized geodesics.
* Example: The metric itself is always a rank-2 Killing tensor; The associated conserved quantity is the norm squared gab uaub = uaua.
* Applications: Integrability of geodesics in Kerr-Newman spacetime.
@ General references: Sommers JMP(73) [and particle constants of motion]; Dolan et al GRG(89) [significance]; Benn JMP(06) [and mechanics]; Coll et al JMP(06)gq [spectral decomposition]; Garfinkle & Glass CQG(10)-a1003 [method, in spacetimes with symmetries]; Cariglia et al CQG(14).
@ Special types of manifolds: Rosquist & Uggla JMP(91) [2D spacetimes]; Smirnov & Yue JMP(04)mp [constant curvature pseudo-Riemannian]; Belgun et al DG&A(06) [symmetric spaces]; Cariglia & Galajinsky PLB(15)-a1503 [Ricci-flat spacetimes]; Vollmer a1602 [in stationary and axisymmetric spacetimes].
@ From conformal Killing vectors: Koutras CQG(92); Barnes et al gq/02-proc; Rani et al CQG(03).
@ Second-rank: Walker & Penrose CMP(70) [Kerr spacetime]; {Bombelli & Rosquist}; Baleanu gq/98, NC-gq/98; Chanu et al JMP(06) [2D flat manifold]; Brink PRD(10); Oota & Yasui IJMPA(10) [generalized Kerr-NUT-de Sitter spacetime]; Keane & Tupper CQG(10) [pp-wave spacetimes].
@ Third-rank: Rosquist & Goliath GRG(98); Karlovini & Rosquist GRG(99)gq/98 [1+1 dimensions]; Baleanu G&C(99).

Killing-Yano Tensors / Forms > s.a. Taub-NUT Metric.
$ Def: An n-th rank Killing-Yano tensor is an n-form ηab... c, such that ∇(m ηa)b... c = 0.
* Relationships: A Killing 2-tensor can be defined from a Killing-Yano tensor by Kmn:= ηmb... c ηnb... c.
* Example: The alternating tensor εab... c is a Killing-Yano tensor; The corresponding rank-2 Killing tensor is (proportional to) the metric.
@ General references: Yano AM(52); Kastor et al CQG(07)-a0705 [conditions for graded Lie algebra with respect to the Schouten-Nijenhuis bracket]; Batista CQG(14)-a1405 [Killing-Yano tensors of order n – 1].
@ In gravitation: Dietz & Rüdiger PRS(81), PRS(82); Hall IJTP(87); Baleanu NCB(99)gq/98 [and Nambu tensors]; Ferrando & Sáez GRG(03) [Rainich problem]; Açık et al GRG(10) [and conserved gravitational currents].
@ Special types of spacetimes: Howarth & Collinson GRG(00) [spherical static]; Jezierski & Lukasik CQG(06) [Kerr]; Kubizňák a0909-proc [black holes]; Acik et al JMP(10) [spherically symmetric]; Houri et al CQG(12) [with torsion, classification]; Garfinkle & Glass JMP(13)-a1302 [spacetimes admitting a hypersurface-orthogonal Killing vector]; > s.a. kerr spacetime.
@ Conformal Killing-Yano tensors: Jezierski APPB(08)-a0705 [asymptotically AdS]; Kubizňák & Krtouš PRD(07)-a0707 [for Plebański-Demiański type-D solutions].

Killing Spinors
* Examples: Manifolds with Killing spinors include nearly Kähler 6-manifolds, nearly parallel G2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds.
@ General references: Baum m.DG/02 [conformal]; Bohle JGP(03) [on Lorentzian manifolds]; in Cariglia CQG(04)ht/03 [and Yano tensors]; Harland & Nölle JHEP(12) [instantons on manifolds with Killing spinors]; Cole & Valiente Kroon CQG(16)-a1601 [implications of their existence].
@ Specific types of spacetimes: Van den Bergh CQG(10)-a0908 [spacetimes admitting non-null valence-two Killing spinors]; Van den Bergh CQG(11) [homogeneous Petrov-type D Killing spinor spacetimes]; Batista PRD(16)-a1512 [in 6D spacetime].
> Online resources: see Wikipedia page.

Other Variations and Generalizations
$ Killing form on a Lie algebra: Given a Lie algebra \(\cal G\), its Killing form is the bilinear form B(X, Y) = tr[Ad(X), Ad(Y)].
@ Generalized Killing tensors: Collinson & Howarth GRG(00); Howe & Lindström JHEP(16)-a1511 [in superspace].


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 27 nov 2016