Conformal Killing vector field: Difference between revisions

In conformal geometry, a conformal Killing vector field (or shortened, a conformal Killing vector) on a manifold of dimension n with (pseudo) Riemannian metric $g$ is a vector field $X$ whose (locally defined) flow defines conformal transformations, i.e. preserve $g$ 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. :${\mathcal {L}}_{X}g=\lambda g$ for some function $\lambda$ on the manifold. For $n\neq 2$ 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 the Killing equation for vector fields that preserve a Riemannian metric.

Densitized metric tensor and Conformal Killing vectors

A vector field $X$ is a Killing vector field iff its flow (locally defined, as it may only be defined for finite times on compact subsets of the manifold) preserves the metric tensor or iff it satisfies ${\mathcal {L}}_{X}g=0$. More generally, define a w-Killing vector field $X$ as a vector field whose flow preserves the densitized metric $g\mu _{g}^{w/n}$, where $\mu _{g}$ is the volume density defined by $g$ (i.e. locally $\mu _{g}={\sqrt {|g|}}dx^{1}\ldots dx^{n}$) and $w\in \mathbf {R}$ is its weight. Note that a Killing vector field preserves $\mu _{g}$ and so automatically also satisfies this more general equation. Also note that $w=-2$ is the unique weight that makes the combination $g\mu _{g}^{w/n}$ invariant under scaling of the metric. Now $X$ is a w-Killing vector field iff

${\mathcal {L}}_{X}\left(g\mu _{g}^{w/n}\right)=({\mathcal {L}}_{X}g)\mu _{g}^{w/n}+{\frac {w}{n}}g\mu _{g}^{(w/n)-1}{\mathcal {L}}_{X}\mu _{g}$.

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

${\mathcal {L}}_{X}g=-{\frac {w}{n}}\mathrm {div} (X)g$.

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

Equivalent formulations

The following are equivalent

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

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 $\lambda =(2/n)\mathrm {div} (X)$.

The conformal Killing equation in (abstract) index notation

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

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

An other index notation to write the conformal Killing equations is

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

See also

• Einstein manifold
• invariant differential operator
• Killing vector field