Math 213a: Complex analysis Problem Set #1 (22 September 2003): Inversions and PSL_{2}(C)

1. [Inversion in higher-dimensional Euclidean space] In Euclidean space R^{n} of arbitrary dimension, inversion I with respect to the sphere of radius R centered at the origin is the map taking any vector x ∈ R^{n} to I(x) := (R^{2}/|x|^{2})x. We take I(0) = ∞ and I(∞) = 0, as in the case n = 2.

i) Show that I preserves hyperspheres (sets of the form {x : |x − x_{0}| = r}), with the convention that a hyperplane is a degenerate special case of a hypersphere, namely a hypersphere passing through ∞.

ii^{∗}) What is the group generated by inversions I and the isometries of R^{n}?

2. Show that the “stereographic projection” from the north pole of a sphere in R^{3} to C can be realized as inversion with respect to a sphere centered at that north pole. Noting that a circle in R^{3} can be obtained as the intersection of two spheres and thus that circles are preserved by inversion, conclude that circles or lines in C correspond to circles on the sphere.

3. [More about the group PGL_{2}(C) acting on the extended complex plane P^{1}(C)] i) Show that any two circles or lines in P^{1}(C) are equivalent under the action of PGL_{2}(C). ii) Show that the subgroup of PGL_{2}(C) preserving a given circle is isomorphic to PGL_{2}(R).

4. i) Show that any fractional linear transformation f such that f(w) = w′ and f(w′) = w for two

distinct complex numbers w, w′ is an involution (i.e., satisfies f(f(z)) = z for all z). ii) Conclude that for any four distinct complex numbers z_{1}, z_{2}, z_{3}, z_{4} there exists a unique fractional

linear transformation f such that f(z_{1}) = z_{2}, f(z_{2}) = z_{1}, f(z_{3}) = z_{4}, and f(z_{4}) = z_{3}.

5. Determine the images under the fractional linear transformations

α : z ↦ \frac{1 + z}{1 − z} , β : z ↦ iz

of the real axis, the imaginary axis, and the unit circle. Show that α and β generate a finite subgroup of PSL_{2}(C) = SL_{2}(C)/{±1}, and interpret its action on P^{1}(C) in terms of the geometry of the Riemann sphere.

6^{∗}. The Hermitian transpose of an n × n complex matrix (a_{ij})_{i,j=1}^{n} is the complex conjugate (ā_{ji}) of its (ordinary) transpose. An invertible n × n matrix is said to be unitary if its inverse is equal to its Hermitian transpose.

i) Show that the unitary n × n complex matrices form a group. ii) Show that an n × n matrix is unitary if and only if it represents a linear transformation: C^{n} → C^{n}

preserving the norm ∥(z_{1}, …, z_{n})∥ = (∑_{i=1}^{n} |z_{i}|^{2})^{1/2}. iii) Show that a 2 × 2 unitary matrix of determinant 1 is a matrix ({a b \atop −b̄ ā}) for some a, b ∈ C with

|a|^{2} + |b|^{2} = 1. (The set of such (a, b) is the unit hypersphere in R^{4}; thus this hypersphere is endowed with a noncommutative group structure.)

iv) Identify P^{1}(C) with the unit sphere in R^{3} by the usual projection, taking the unit circle of C to the equator of the sphere. Show that the group of orientation-preserving isometries of this sphere is then identified with the quotient of the group SU_{2}(C) of 2 × 2 unitary matrices of determinant 1 by its normal subgroup {±1}.

This problem set is due Monday, September 29, at the beginning of class.