Killing vector field - Definition and Overview

In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. Killing fields are named for Wilhelm Killing.

Specifically, a vector field X is a Killing field if the Lie derivative with respect to X of the metric g vanishes:

<math>\mathcal{L}_X g = 0<math>

In terms of the covariant derivative, this is

<math>g(\nabla_Y X, Z) + g(Y, \nabla_Z X) = 0<math>

for all vectors Y and Z. In local coordinates, this amounts to the equation

<math>\nabla_i X_j + \nabla_j X_i = 0<math>

A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all covariant derivatives of the field at the point).

The Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. This is the Lie algebra of the isometry group of the manifold.

For compact manifolds

• Negative Ricci curvature implies there are no nontrivial (nonzero) Killing fields.
• Nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector field is identically zero.
• If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero.