Conformal Killing vector field: Difference between revisions

In conformal geometry, the conformal Killing equation on a manifold of space-dimension n with Riemannian metric $g$ describes those vector fields $X$ which preserve $g$ up to scale, i.e.

${\mathcal {L}}_{X}g=\lambda g$

for some function $\lambda$ (where ${\mathcal {L}}_{X}$ is the Lie derivative). Vector fields that satisfy the conformal Killing equation are exactly those vector fields whose flow preserves the conformal structure of the manifold. The name Killing refers to Wilhelm Killing, who first investigated the Killing equation for vector fields that preserve a Riemannian metric.

By taking the trace we find that necessarily $\lambda ={\frac {2}{n}}\mathrm {div} X$. Therefore we can write the conformal Killing equation as

$\left({\mathcal {L}}_{X}-{\frac {2\,\mathrm {div} \,X}{n}}\right)g=0.$

In abstract indices,

$\nabla _{(a}X_{b)}-{\frac {2}{n}}g_{ab}\nabla _{c}X^{c}=0,$

where the round brackets denote symmetrization and $\nabla$ is the covariant derivative. In differential geometry there are many different notations in use and another way to write the conformal Killing equations is

$X_{i;k}+X_{k;i}-{\frac {2}{n}}g_{ik}X_{;j}^{j}=0,$

For any n but 2, there is a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions.

See also

• Einstein manifold
• invariant differential operator