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Conformal equivalence
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#REDIRECT [[Conformal geometry]] [[File:Riemann sphere1.svg|thumb|300px|[[Stereographic projection]] is a conformal equivalence between a portion of the [[sphere]] (with its standard metric) and the [[plane (geometry)|plane]] with the metric <math> \frac{4}{(1 + X^2 + Y^2)^2} \; ( dX^2 + dY^2)</math>.|right]] In [[mathematics]] and [[theoretical physics]], two [[Thurston model geometry|geometries]] are '''conformally equivalent''' if there exists a [[conformal transformation]] (an angle-preserving transformation) that maps one geometry to the other one.<ref>{{citation|title=Functions of One Complex Variable II|series=[[Graduate Texts in Mathematics]]|volume=159|first=John B.|last=Conway|publisher=Springer|year=1995|isbn=9780387944609|page=29|url=https://books.google.com/books?id=JN0hz3qO1eMC&pg=PA29}}.</ref> More generally, two [[Riemannian metric]]s on a [[manifold]] ''M'' are conformally equivalent if one is obtained from the other by multiplication by a positive function on ''M''.<ref>{{citation|title=Global Calculus|first=S.|last=Ramanan|publisher=American Mathematical Society|isbn=9780821872406|year=2005|page=221|url=https://books.google.com/books?id=1INoRKtgndcC&pg=PA221}}.</ref> Conformal equivalence is an [[equivalence relation]] on geometries or on Riemannian metrics. == See also == * [[conformal geometry]] * [[biholomorphy|biholomorphic equivalence]] * [[AdS/CFT correspondence]] ==References== {{reflist}} [[Category:Conformal geometry]]
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