Killing field

Let $(𝑀, 𝑔)$ be a semi-Riemannian $𝐶 𝛼$ -manifold. A Killing field $𝜉 𝑎 \in 𝔦 𝔰 𝔬 (𝑀, 𝑔)$ is a vector field which generates a flow which is an isometry. #m/def/geo/diff Equivalently, the Lie derivative of the metric tensor $𝑔 𝑎 𝑏$ along $𝜉 𝑎$ vanishes

L 𝜉 𝑔 𝑎 𝑏 = 0,

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

\nabla (𝑎 𝜉 𝑏) = 0 .

Proof of equivalence

#missing/proof

The space $𝔦 𝔰 𝔬 (𝑀, 𝑔)$ of all Killing fields form a Lie subalgebra of $𝔛 (𝑀)$ , with the corresponding Lie group being the isometries $I s o (𝑀, 𝑔)$ .

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