In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric
(also called a conformal Killing vector, CKV, or conformal colineation), is a vector field
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.
there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions.
is a Killing vector field if and only if its flow preserves the metric tensor
(strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time).
More generally, define a w-Killing vector field
as a vector field whose (local) flow preserves the densitized metric
Note that a Killing vector field preserves
and so automatically also satisfies this more general equation.
is the unique weight that makes the combination
Therefore, in this case, the condition depends only on the conformal structure.
this is equivalent to Taking traces of both sides, we conclude
and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric.
has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.
However, the last two forms are also equivalent: taking traces shows that necessarily
The last form makes it clear that any Killing vector is also a conformal Killing vector, with
(aka associated covariant vector aka vector with lowered indices), 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 In
-dimensional flat space, that is Euclidean space or pseudo-Euclidean space, there exist globally flat coordinates in which we have a constant metric
In these coordinates, the connection components vanish, so the covariant derivative is the coordinate derivative.
The conformal Killing equation in flat space is
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.
more generators, known as special conformal transformations, given by where the traceless part of
For convenience we rewrite the conformal Killing equation as
) Applying an extra derivative, relabelling indices and taking a linear combination of the resulting equations gives
A combination of derivatives of this and the original conformal Killing equation gives
is at most quadratic in coordinates, with general form