23 - Conformal groups

Summary

Our concern in this chapter is with the group Conf(X) of conformal transformations of a non-degenerate real quadratic space X of finite dimension n and signature (p, q), and with a description of such groups that involves Clifford algebras. In doing so we shall draw heavily on Chapter 18 which was concerned with the study of 2 × 2 Clifford matrices.

Let X and Y be finite-dimensional quadratic spaces and f : X ↣ Y a smooth map. Then f is said to be conformal if the differential dfx of f at any point x is of the form p(x)t, where p(x) is a non-zero real number and t : X→ Y is an orthogonal map, and so is such that, for any u, v ∈ X, dfx(u) · dfx(v) = (p(x))²u · v; that is it is conformal if it preserves angles. More generally, let X and Y be finite-dimensional smooth manifolds and f : X ↣ Y a smooth map. Then f is said to be conformal if the differential dfx of f at any point x of X is a non-zero real multiple of an orthogonal map.

It is well-known that any holomorphic map f : C ↣ C, with C identified as a quadratic space with R² with its standard scalar product, is conformal. Conformal transformations of quadratic spaces of dimension greater than 2 are more restricted, as follows, in the positive-definite case at least, from a theorem of Liouville (1850).