Definition:Killing Equation

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 $Z$.