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In [[mathematics]], the '''conformal group''' is the [[group (mathematics)|group]] of transformations from a space to itself that preserve all angles within the space. More formally, it is the group of transformations that preserve the [[conformal geometry]] of the space. Several specific conformal groups are particularly important:
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* The conformal [[orthogonal group]]. If ''V'' is a vector space with a [[quadratic form]] ''Q'', then the conformal orthogonal group CO(''V'',''Q'') is the group of linear transformations ''T'' of ''V'' such that for all ''x'' in ''V'' there exists a scalar &lambda; such that
::<math>Q(Tx) = \lambda^2 Q(x)</math>
:The conformal orthogonal group is equal to the [[orthogonal group]] times the group of [[dilation]]s{{dn|date=April 2012}}.
 
* The conformal group of the sphere. The group of conformal transformations of the [[n-sphere]] is generated by the [[inversive geometry|inversions in circles]]. This group is also known as the [[Möbius group]].
 
All conformal groups are Lie Groups.
 
==References==
* {{cite book | first = S. | last = Kobayashi | title = Transformation Groups in Differential Geometry | series = Classics in Mathematics | publisher = Springer | year = 1972 | isbn = 3-540-58659-8 | oclc = 31374337}}
* {{citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | id = ISBN 0-387-94732-9}}.
 
{{algebra-stub}}
[[Category:Conformal geometry]]

Latest revision as of 01:09, 27 November 2014

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