# nLab Killing vector field

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

A Killing vector on a (pseudo-)Riemannian manifold is equivalently

Similarly a Killing spinor is a covariantly constant spinor.

## Definition

For $(X,g)$ a Riemannian manifold (or pseudo-Riemannian manifold) a vector field $v \in \Gamma(T X)$ is called a Killing vector field if it generates isometries of the metric $g$. More precisely, if, equivalently

• the Lie derivative of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;

• the flow $\exp(v) : X \times \mathbb{R} \to X$ is a flow by isometries.

## Properties

The flows of Killing vectors are isometries of the Riemannian manifold onto itself.

Revised on February 19, 2015 15:37:31 by David Corfield (129.12.18.141)