Conformal group

In mathematics, the conformal group is the 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:

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 λ such that
$Q(Tx) = \lambda^2 Q(x)$

The conformal orthogonal group is equal to the orthogonal group times the group of dilations.

The conformal group of the sphere. The group of conformal transformations of the n-sphere is generated by the inversions in circles. This group is also known as the Möbius group. When applied to electrodynamics it is related to the conformal group of spherical wave transformations.

All conformal groups are Lie Groups.

References

Kobayashi, S. (1972). Transformation Groups in Differential Geometry. Classics in Mathematics. Springer. ISBN 3-540-58659-8. OCLC 31374337.
Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9 .