nLab
Killing-Yano tensor

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

On a (pseudo-)Riemannian manifold (X,g)(X,g) a Killing-Yano tensor is a differential form ω\omega such that

(μω α 0),α 1,,α n=0, \nabla_{(\mu} \omega_{\alpha_0), \alpha_1, \cdots , \alpha_n} = 0 \,,

where \nabla is the covariant derivative with respect to the Levi-Civita connection of gg.

There is also a variant of conformal Killing-Yano tensors

(…)

Properties

Killing-Yano tensors serve as “square roots” of Killing tensor. In a spacetime with a Killing tensor HH the relativistic particle has an extra conserved quantity. If it refines to a Killing-Yano tensor then also the spinning particle or superparticle has an extra odd conserved quantity. If H=gH = g then this is an extra worldline supersymmetry.

Examples

The Kerr spacetime admits a conformal Killing-Yano tensor (…)

References

For instance

  • O. P. Santillan, Killing-Yano tensors and some applications (arXiv:1108.0149)

  • Jacek Jezierski, Maciej Łukasik, Conformal Yano-Killing tensor for the Kerr metric and conserved quantities (arXiv:gr-qc/0510058)

  • W. Dietz and R. Rüdiger, Space-Times Admitting Killing-Yano Tensors. I Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 375, No. 1762 (Mar. 31, 1981), pp. 361 (JSTOR)

Revised on September 17, 2011 13:23:18 by Urs Schreiber (82.113.99.0)