Killing vector field

Summary

A Killing vector field on a pseudo-Riemannian manifold is a vector field along which the metric does not change. That is, the Lie derivative of the metric along it is zero. Its local flow consists of isometries, so Killing fields are the infinitesimal symmetries of the metric. They form a Lie algebra, which is the Lie algebra of the isometry group when the manifold is complete. In physics they give conserved quantities along geodesics.