Definition:Killing Equation
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields of $M$.
Suppose $Z \in \map {\mathfrak{X}} M$ is a smooth vector field such that:
- $\ds \forall X, Y \in \map {\mathfrak{X}} M : \map g {\nabla_X Z, Y} + \map g {X, \nabla_Y Z} = 0$
where $\nabla_X Z$ denotes the covariant derivative of $Z$ along $X$.
Then the equation above is the Killing equation for $Z$.
Source of Name
This entry was named for Wilhelm Karl Joseph Killing.
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