Conformal Killing vector field


In conformal geometry, a conformal Killing vector field on a manifold of dimension n with Riemannian metric , is a vector field whose flow defines conformal transformations, i.e. preserve 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.
for some function on the manifold. For 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.

Densitized metric tensor and Conformal Killing vectors

A vector field is a Killing vector field iff its flow preserves the metric tensor or iff it satisfies
More generally, define a w-Killing vector field as a vector field whose flow preserves the densitized metric, where is the volume density defined by and is its weight. Note that a Killing vector field preserves and so automatically also satisfies this more general equation. Also note that is the unique weight that makes the combination invariant under scaling of the metric, therefore, only depending on the conformal structure.
Now
Since this is equivalent to
Taking traces of both sides, we conclude. Hence for, necessarily and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for, the flow of merely has to 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. The flow of preserves the conformal structure,
  5. for some function
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.

The conformal Killing equation in (abstract) index notation

Using that where is the Levi Civita derivative of , and is the dual 1 form of , and 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