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)

Last revised on September 17, 2011 at 13:23:18. See the history of this page for a list of all contributions to it.