In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, or conformal colineation), is a vector field whose (locally defined) 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 equa