Conformal equivalence

1. REDIRECT Conformal geometry

File:Riemann sphere1.svg is a conformal equivalence between a portion of the sphere (with its standard metric) and the plane with the metric $\backslash frac\{4\}\{(1\; +\; X^2\; +\; Y^2)^2\}\; \backslash ;\; (\; dX^2\; +\; dY^2)$.|right]]

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.{{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}}.

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.{{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}}. Conformal equivalence is an equivalence relation on geometries or on Riemannian metrics.