Conformal Map - Riemannian Geometry

Riemannian Geometry

In Riemannian geometry, two Riemannian metrics and on smooth manifold are called conformally equivalent if for some positive function on . The function is called the conformal factor.

A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map.

One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.