Conformal equivalence

Stereographic projection is a conformal equivalence between a portion of the sphere (with its standard metric) and the plane with the metric $\frac{4}{(1 + X^2 + Y^2)^2} \; ( dX^2 + dY^2)$.

In mathematics and theoretical physics, two geometries are conformally equivalent if there exists a conformal transformation (an angle-preserving transformation) that maps one geometry to the other one. More generally, two Riemannian metrics on a manifold M are conformally equivalent if one is obtained from the other by multiplication by a positive function on M. Conformal equivalence is an equivalence relation on geometries or on Riemannian metrics.

See also

• conformal geometry
• biholomorphic equivalence
• AdS/CFT correspondence