Killing vector field

Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold.

More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.

is a Killing vector field if the Lie derivative with respect to

In local coordinates, this amounts to the Killing equation[2] This condition is expressed in covariant form.

A toy example for a Killing vector field is on the upper half-plane

is typically called the hyperbolic plane and has Killing vector field

transports the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis).

The isometry group of the upper half-plane model (or rather, the component connected to the identity) is

(see Poincaré half-plane model), and the other two Killing fields may be derived from considering the action of the generators of

That is, by using the a priori knowledge that spheres can be embedded in Euclidean space, it is immediately possible to guess the form of the Killing fields.

gives the standard metric on the sphere, Intuitively, a rotation about any axis should be an isometry.

Any transformation that moves points closer or farther apart cannot be an isometry; therefore, the generator of such motion cannot be a Killing field.

These three Killing fields form a complete set of generators for the algebra.

Together with space-time translations, this forms the Lie algebra for the Poincaré group.

Here we derive the Killing fields for general flat space.

is the Riemann curvature tensor, the following identity may be proven for a Killing field

⁠, we get a basis for the generalised Poincaré algebra of isometries of flat space: These generate pseudo-rotations (rotations and boosts) and translations respectively.

Heuristically, we can derive the dimension of the Killing field algebra.

The Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. This is the Lie algebra of the isometry group of the manifold if M is complete.

A Riemannian manifold with a transitive group of isometries is a homogeneous space.

For compact manifolds The covariant divergence of every Killing vector field vanishes.

is conserved: This aids in analytically studying motions in a spacetime with symmetries.

⁠, which are properties typical of a stress-energy tensor, and a Killing vector ⁠

of the total space, and the Killing fields show how to "slide along" that submanifold.

Yet, in general, the number of Killing fields is larger than the dimension of that tangent space.

⁠; the remaining degenerate linear combinations define an orthogonal space ⁠

The Cartan involution is defined as the mirroring or reversal of the direction of a geodesic.

When restricted to geodesics along the Killing fields, it is also clearly an isometry.

As these are even and odd parity subspaces, the Lie brackets split, so that

[citation needed] For the special case of a symmetric space, one explicitly has that ⁠