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In conformal geometry, a conformal Killing vector field on a manifold of dimension n with Riemannian metric g {\displaystyle g} , is a vector field X {\displaystyle X} whose flow defines conformal transformations, that is, preserve g {\displaystyle g} 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. L X g = λ g {\displaystyle {\mathcal {L}}_{X}g=\lambda g} for some function λ {\displaystyle \lambda } on the manifold. For n ≠ 2 {\displaystyle n\neq 2} 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.