# Killing vector field

A Killing vector field (red) with integral curves (blue) on a sphere.

In mathematics, a Killing vector field (often just Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-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. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.

## Definition

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

${\displaystyle {\mathcal {L}}_{X}g=0\,.}$

In terms of the Levi-Civita connection, this is

${\displaystyle g(\nabla _{Y}X,Z)+g(Y,\nabla _{Z}X)=0\,}$

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

${\displaystyle \nabla _{\mu }X_{\nu }+\nabla _{\nu }X_{\mu }=0\,.}$

This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.

## Examples

• The vector field on a circle that points clockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.

## Properties

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 if M is complete.

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 j field is identically zero.
• If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero.

The divergence of every Killing vector field vanishes.

### Geodesics

Each Killing vector corresponds to a quantity which is conserved along geodesics. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. That is, along a geodesic with some affine parameter ${\displaystyle \lambda }$, the equation ${\displaystyle {\frac {d}{d\lambda }}(K_{\mu }{\frac {dx^{\mu }}{d\lambda }})=0}$ is satisfied. This aids in analytically studying motions in a spacetime with symmetries.[2]

## Notes

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## References

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