Killing vector field

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 generator of isometries; that is, flow 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.

Explanation

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

<math>mathcal_ g = 0 ,.</math>

In terms of the Levi-Civita connection, this is

<math>g(nabla_ X, Z) + g(Y, nabla_ X) = 0 ,</math>

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

<math>nabla_ X_ + nabla_ X_ = 0 ,.</math>

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.

• If the metric coefficients <math>g_ ,</math> in some coordinate basis <math>dx^ ,</math> are......
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