nLab
Killing vector field

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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

  • the Lie derivative of gg along vv vanishes: vg=0\mathcal{L}_v g = 0;

  • the flow exp(v):X×X\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)