Definition:Killing Vector Field

Definition
Let $\struct {M, g}$ be a Riemannian manifold.

Let $\map {\mathfrak X} M$ be the space of smooth vector fields of $M$.

Let $X \in \map {\mathfrak X} M$.

Let $\LL$ be the Lie derivative.

Suppose $\map {\LL_X} g = 0$.

Then $X$ is called the Killing vector field.