Differential geometry MOC

Killing field

Let (𝑀,𝑔) be a [[Semi-Riemannian manifold|semi-Riemannian 𝐶𝛼-manifold]]. A Killing field 𝜉𝑎 ∈𝔦𝔰𝔬(𝑀,𝑔) is a vector field which generates a flow which is an isometry. diff Equivalently, the Lie derivative of the metric tensor 𝑔𝑎𝑏 along 𝜉𝑎 vanishes

L𝜉𝑔𝑎𝑏=0,

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

∇(𝑎𝜉𝑏)=0.

Proof of equivalence

proof

The space 𝔦𝔰𝔬(𝑀,𝑔) of all Killing fields form a Lie subalgebra of 𝔛(𝑀), with the corresponding Lie group being the isometries Iso⁡(𝑀,𝑔).

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