In conformal geometry

Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...

, the conformal Killing equation on a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 of space-dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

 n with metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

  describes those vector fields which preserve up to scale, i.e.
for some function (where is the Lie derivative

Lie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

). Vector fields that satisfy the conformal Killing equation are exactly those vector fields whose flow

Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...

 preserves the conformal structure of the manifold. The name Killing refers to Wilhelm Killing

Wilhelm Killing

Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....

, who first investigated the Killing equation for vector fields that preserve a Riemannian metric.

By taking the trace we find that necessarily . Therefore we can write the conformal Killing equation as
In abstract indices
where the round brackets denote symmetrization.