# nLab Killing tensor

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

For $(X,g)$ a (pseudo-)Riemannian manifold a Killing tensor is a section of a symmetric power of the tangent bundle

$K \in Sym^k \Gamma(T X)$

which is covariantly constant in that

$\nabla_{(\mu} K_{\alpha_1, \cdots, \alpha_k)} = 0 \,.$

For $k = 1$ this reduces to the notion of Killing vector.

## Properties

For every Killing tensor $K$ on $(X,g)$ the dynamics of the relativistic particle on $X$ has a further conserved quantity. In the canonical case $K = g$ this quantity is the Hamiltonian of the particle (in the case of a relativistic particle its four-velocity normalization).

The analog of this for spinning particles and superparticles are Killing-Yano tensors.

Revised on April 2, 2015 13:52:01 by Tim Porter (2.31.48.51)