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• Main Author: Joel H. Shapiro
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Subjects: North Pole
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.541.7106; http://www.mth.msu.edu/~shapiro/pubvit/downloads/rs_rotation/rotation.pdf

Abstract. 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 “legit-imizing ” 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 C ̂ onto the Riemann Sphere S2 via the stereographic projection, making ∞ correspond to the north pole. Recall also that the stereographic projection S2\{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 z ̂ 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+bcz+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