# nLab Killing-Yano tensor

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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 $(X,g)$ a Killing-Yano tensor is a differential form $\omega$ such that

$\nabla_{(\mu} \omega_{\alpha_0), \alpha_1, \cdots , \alpha_n} = 0 \,,$

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

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 $H$ 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 $H = g$ 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.