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High Quality Content by WIKIPEDIA articles! In conformal geometry, the conformal Killing equation on a manifold of space-dimension n with metric g describes those vector fields X which preserve g up to scale, i.e. In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces. In more than two dimensions, conformal geometry may refer either to the study of conformal transformations of "flat" spaces (such as Euclidean spaces or spheres), or, more commonly, to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry.