(From Wikipedia)
In mathematics, 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:
$$ \mathcal{L}_X g=0 $$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.
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Author of the notes: Antonio J. Pan-Collantes
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