In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric whose (locally defined) flow defines conformal transformations up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields.

Densitized metric tensor and Conformal Killing vectors

A vector field is a Killing vector field (strictly speaking for each compact is Killing if and only if it satisfies

is the Lie derivative.

More generally, define a w invariant under scaling of the metric. Therefore, in this case, the condition depends only on the conformal structure is a w-Killing vector field if and only if

this is equivalent to

and a w has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.

Equivalent formulations

The following are equivalent

1. is a conformal Killing vector field,

2. preserves the conformal structure,

.

The conformal Killing equation

(aka covariant derivative is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as

Another index notation to write the conformal Killing equations is

Examples

Flat space

-dimensional flat space, that is Euclidean space or pseudo-Euclidean space The solutions to the flat space conformal Killing equation includes the solutions to the flat space Killing equation discussed in the article on Killing vector fields. These generate the Poincaré group of isometries of flat space. Considering the ansatz is symmetric. It follows that this is a dilatation.

more generators, known as special conformal transformations, given by

. special conformal transformations comprise the conformal algebra, which generate the conformal group of pseudo-Euclidean space.

See also

• Affine vector field

• Conformal Killing tensor

• Curvature collineation

• Einstein manifold

• Homothetic vector field

• Invariant differential operator

• Killing vector field

• Matter collineation

• Spacetime symmetries

References

Further reading

• Wald, R. M. (1984). General Relativity. The University of Chicago Press.