Conformal Killing vector field

In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric ${\displaystyle g}$ (also called a conformal Killing vector, or conformal colineation), is a vector field ${\displaystyle X}$ whose (locally defined) flow defines conformal transformations, i.e. preserve ${\displaystyle g}$ up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g. ${\displaystyle {\mathcal {L}}_{X}g=\lambda g}$ for some function ${\displaystyle \lambda }$ on the manifold. For ${\displaystyle n\neq 2}$ 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 that preserve a Riemannian metric and satisfy the Killing equation ${\displaystyle {\mathcal {L}}_{X}g=0}$.

Densitized metric tensor and Conformal Killing vectors

A vector field ${\displaystyle X}$ is a Killing vector field iff its flow preserves the metric tensor ${\displaystyle g}$ (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). This can be formulated infinitesimally (and more conveniently) as ${\displaystyle X}$ is Killing iff it satisfies

${\displaystyle {\mathcal {L}}_{X}g=0.}$

where ${\displaystyle {\mathcal {L}}_{X}}$ is the Lie derivative.

More generally, define a w-Killing vector field ${\displaystyle X}$ as a vector field whose (local) flow preserves the densitised metric ${\displaystyle g\mu _{g}^{w}}$, where ${\displaystyle \mu _{g}}$ is the volume density defined by ${\displaystyle g}$ (i.e. locally ${\displaystyle \mu _{g}={\sqrt {|\det(g)|}}\,dx^{1}\cdots dx^{n}}$) and ${\displaystyle w\in \mathbf {R} }$ is its weight. Note that a Killing vector field preserves ${\displaystyle \mu _{g}}$ and so automatically also satisfies this more general equation. Also note that ${\displaystyle w=-2/n}$ is the unique weight that makes the combination ${\displaystyle g\mu _{g}^{w}}$ invariant under scaling of the metric, therefore, in this case, the condition only depending on the conformal structure. Now ${\displaystyle X}$ is a w-Killing vector field iff

${\displaystyle {\mathcal {L}}_{X}\left(g\mu _{g}^{w}\right)=({\mathcal {L}}_{X}g)\mu _{g}^{w}+wg\mu _{g}^{w-1}{\mathcal {L}}_{X}\mu _{g}=0.}$

Since ${\displaystyle {\mathcal {L}}_{X}\mu _{g}=\operatorname {div} (X)\mu _{g}}$ this is equivalent to

${\displaystyle {\mathcal {L}}_{X}g=-w\operatorname {div} (X)g}$.

Taking traces of both sides, we conclude ${\displaystyle 2\mathop {\mathrm {div} } (X)=-wn\operatorname {div} (X)}$. Hence for ${\displaystyle w\neq -2/n}$, necessarily ${\displaystyle \operatorname {div} (X)=0}$ and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for ${\displaystyle w=-2/n}$, the flow of ${\displaystyle X}$ merely has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.

Equivalent formulations

The following are equivalent

1. ${\displaystyle X}$ is a conformal Killing vector field,
2. The (locally defined) flow of ${\displaystyle X}$ preserves the conformal structure,
3. ${\displaystyle {\mathcal {L}}_{X}(g\mu _{g}^{-2/n})=0,}$
4. ${\displaystyle {\mathcal {L}}_{X}g={\frac {2}{n}}\operatorname {div} (X)g,}$
5. ${\displaystyle {\mathcal {L}}_{X}g=\lambda g}$ for some function ${\displaystyle \lambda .}$

The discussion above proves the equivalence of all but the seemingly more general last form. However, the last two forms are also equivalent: taking traces shows that necessarily ${\displaystyle \lambda =(2/n)\operatorname {div} (X)}$.

The conformal Killing equation in (abstract) index notation

Using that ${\displaystyle {\mathcal {L}}_{X}g=2\left(\nabla X^{\flat }\right)^{\mathrm {symm} }}$ where ${\displaystyle \nabla }$ is the Levi Civita derivative of ${\displaystyle g}$ (aka covariant derivative), and ${\displaystyle X^{\flat }=g(X,\cdot )}$ is the dual 1 form of ${\displaystyle X}$ (aka associated covariant vector aka vector with lowered indices), and ${\displaystyle {}^{\mathrm {symm} }}$ is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as

${\displaystyle \nabla _{a}X_{b}+\nabla _{b}X_{a}={\frac {2}{n}}g_{ab}\nabla _{c}X^{c}.}$

Another index notation to write the conformal Killing equations is

${\displaystyle X_{\mu$ ;\nu }+X_{\nu ;\mu }={\frac {2}{n}}g_{\mu \nu }X_{;\sigma }^{\sigma }.}

See also

• Affine vector field
• Curvature collineation
• Einstein manifold
• Homothetic vector field
• invariant differential operator
• Killing vector field
• Matter collineation
• Spacetime symmetries