In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.

Specifically, a vector field ${\displaystyle X}$ is a Killing vector field if the Lie derivative with respect to ${\displaystyle X}$ of the metric tensor ${\displaystyle g}$ vanishes:

In terms of the Levi-Civita connection, this is

for all vectors ${\displaystyle Y}$ and ⁠${\displaystyle Z}$⁠. In local coordinates, this amounts to the Killing equation

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.

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

A toy example for a Killing vector field is on the upper half-plane ${\displaystyle M=\mathbb {R} _{y>0}^{2}}$ equipped with the Poincaré metric ⁠${\displaystyle g=y^{-2}\left(dx^{2}+dy^{2}\right)}$⁠. The pair ${\displaystyle (M,g)}$ is typically called the hyperbolic plane and has Killing vector field ${\displaystyle \partial _{x}}$ (using standard coordinates). This should be intuitively clear since the covariant derivative ${\displaystyle \nabla _{\partial _{x}}g}$ transports the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis).

Furthermore, the metric tensor is independent of ${\displaystyle x}$ from which we can immediately conclude that ${\displaystyle \partial _{x}}$ is a Killing field using one of the results below in this article.