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.

Source of Name

This entry was named for Wilhelm Karl Joseph Killing.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Problems