In
mathematics, the conformal group of an
inner product space is the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the
conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
of the space.
Several specific conformal groups are particularly important:
* The conformal
orthogonal group. If ''V'' is a vector space with a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
''Q'', then the conformal orthogonal group is the group of linear transformations ''T'' of ''V'' for which there exists a scalar ''λ'' such that for all ''x'' in ''V''
*:
:For a
definite quadratic form
In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical ...
, the conformal orthogonal group is equal to the
orthogonal group times the group of
dilations.
* The conformal group of the
sphere is generated by the
inversions in circles. This group is also known as the
Möbius group.
* In
Euclidean space E
''n'', , the conformal group is generated by inversions in
hyperspheres.
* In a
pseudo-Euclidean space E
''p'',''q'', the conformal group is .
All conformal groups are
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s.
Angle analysis
In Euclidean geometry one can expect the standard circular
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
to be characteristic, but in
pseudo-Euclidean space there is also the
hyperbolic angle. In the study of
special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by
rapidity, a hyperbolic angle. One way to describe a
Lorentz boost is as a
hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are
conformal transformations with respect to the hyperbolic angle.
A method to generate an appropriate conformal group is to mimic the steps of the
Möbius group as the conformal group of the ordinary
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. Pseudo-Euclidean geometry is supported by alternative complex planes where points are
split-complex numbers or
dual numbers. Just as the Möbius group requires the
Riemann sphere, a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by
linear fractional transformations on the appropriate plane.
Mathematical definition
Given a (
Pseudo-)
Riemannian manifold with
conformal class
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...