Conformal equivalence

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Conformal geometry

{\displaystyle {\frac {4}{(1+X^{2}+Y^{2})^{2}}}\;(dX^{2}+dY^{2})} — 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.^[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.

References

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