Which Linear-fractional Transformations Induce Rotations of the Sphere?

These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course in complex analysis. The goal is to prove that a mapping of the Riemann sphere to itself is a rotation if and only if the corresponding map induced on the plane by stereographic projection is a linear fractional whose (two-by-two) coefficient matrix is unitary. 1. Spheres, points, and subspaces 1.1. Point at infinity. Recall that we have discussed two ways of “legitimizing” the “point at infinity” for the complex plane: (a) The Riemann Sphere S2 (cf. our textbook [S, pp. 8–11]). Here the idea is to map the extended plane Ĉ onto the Riemann Sphere S2 via the stereographic projection , making∞ correspond to the north pole. Recall also that the stereographic projection S\{North Pole} → C is conformal. (b) Complex Projective space CP1 (cf. [S, p. 25]). We regard this as the collection of one dimensional subspaces of C2, with the point z ∈ C identified with the subspace spanned by the column vector [z, 1]t (where the superscript “t” denotes “transpose”), and ∞ identified with the one spanned by [1, 0]t. 1.2. Notation. Let ẑ denote the one dimensional subspace of C2 spanned by the vector [z, 1]t if z ∈ C, and let ∞̂ be the subspace spanned by [1, 0]t. 1.3. Matrices and LFT’s. We have also made a connection between linear fractional transformations and matrices. This begins in a purely formal way by associating each LFT φ(z) = az+b cz+d with the two-by-two complex nonsingular matrix [φ] = [ a b c d ] , noting that actually [φ] should not just stand for one matrix, but for the one-parameter family: all the multiples Date: March 19, 2004.