Killing field

Let be a semi-Riemannian -manifold. A Killing field is a vector field which generates a flow which is an isometry. #m/def/geo/diff Equivalently, the Lie derivative of the metric tensor along vanishes

or the symmetrization of the covariant derivative by the Levi-Civita connexion vanishes

Proof of equivalence

#missing/proof

The space of all Killing fields form a Lie subalgebra of , with the corresponding Lie group being the isometries .

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