Math 213a: Complex analysis
Problem Set #1 (22 September 2003):
Inversions and PSL2(C)
1. [Inversion in higher-dimensional Euclidean space] In Euclidean space Rnof arbitrary dimension,
inversion Iwith respect to the sphere of radius Rcentered at the origin is the map taking
any vector x∈Rnto I(x) := (R2/|x|2)x. We take I(0) = ∞and I(∞) = 0, as in the case n= 2.
i) Show that Ipreserves hyperspheres (sets of the form {x:|x−x0|=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 Iand the isometries of Rn?
2. Show that the “stereographic projection” from the north pole of a sphere in R3to Ccan be realized
as inversion with respect to a sphere centered at that north pole. Noting that a circle in R3can
be obtained as the intersection of two spheres and thus that circles are preserved by inversion,
conclude that circles or lines in Ccorrespond to circles on the sphere.
3. [More about the group PGL2(C) acting on the extended complex plane P1(C)]
i) Show that any two circles or lines in P1(C) are equivalent under the action of PGL2(C).
ii) Show that the subgroup of PGL2(C) preserving a given circle is isomorphic to PGL2(R).
4. i) Show that any fractional linear transformation fsuch that f(w) = w0and f(w0) = wfor two
distinct complex numbers w, w0is an involution (i.e., satisfies f(f(z)) = zfor all z).
ii) Conclude that for any four distinct complex numbers z1, z2, z3, z4there exists a unique fractional
linear transformation fsuch that f(z1) = z2,f(z2) = z1,f(z3) = z4, and f(z4) = z3.
5. Determine the images under the fractional linear transformations
α:z7→ 1 + z
1−z, β :z7→ iz
of the real axis, the imaginary axis, and the unit circle. Show that αand βgenerate a finite
subgroup of PSL2(C) = SL2(C)/{±1}, and interpret its action on P1(C) in terms of the geometry
of the Riemann sphere.
6∗. The Hermitian transpose of an n×ncomplex matrix (aij )n
i,j=1 is the complex conjugate (¯aji) of
its (ordinary) transpose. An invertible n×nmatrix is said to be unitary if its inverse is equal to
its Hermitian transpose.
i) Show that the unitary n×ncomplex matrices form a group.
ii) Show that an n×nmatrix is unitary if and only if it represents a linear transformation: Cn→Cn
preserving the norm k(z1, . . . , zn)k= (Pn
i=1 |zi|2)1/2.
iii) Show that a 2 ×2 unitary matrix of determinant 1 is a matrix ( a b
−¯
b¯a) for some a, b ∈Cwith
|a|2+|b|2= 1. (The set of such (a, b) is the unit hypersphere in R4; thus this hypersphere is
endowed with a noncommutative group structure.)
iv) Identify P1(C) with the unit sphere in R3by the usual projection, taking the unit circle of Cto
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 SU2(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.