Conformal equivalence

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• Conformal geometry

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

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.[1] 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.[2] Conformal equivalence is an equivalence relation on geometries or on Riemannian metrics.

See also

• conformal geometry
• biholomorphic equivalence
• AdS/CFT correspondence

References

1. Conway, John B. (1995), Functions of One Complex Variable II, Graduate Texts in Mathematics, vol. 159, Springer, p. 29, ISBN 9780387944609.
2. Ramanan, S. (2005), Global Calculus, American Mathematical Society, p. 221, ISBN 9780821872406.