nLab
Killing vector field

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

A Killing vector field is an infinitesimal isometry.

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.

Created on October 1, 2010 12:29:54 by Urs Schreiber (89.204.153.98)