Killing vector field

(From Wikipedia)
A Killing vector field (often called a 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.

They can be defined in terms of the Lie derivative: let's call $X$ to a vector field on a pseudo-Riemannian manifold $(M,g)$. It is a Killing vector field if:

The Lie bracket of two Killing fields is again a Killing field, so they constitute a Lie algebra contained in $\mathfrak{X}(M)$. It is associated to the isometry group of the manifold, if $M$ is complete.

A Killing vector field is a Jacobi field for any geodesic (see this answer MSE).