# 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 energy Hamiltonian of the particle.

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

Revised on February 3, 2015 22:46:20 by Anonymous Coward (149.144.211.93)