The action of $PGL(2,{\Bbb C})$ on ${\Bbb C}P^1$

From the algebraic viewpoint, there is little difference between the description of the action of the complex projective group on complex projective space and the description of the action of the real projective group on real projective space.

Exercise 2.1 (00)   Show that the action of $GL(n+1,{\Bbb C})$ on ${\Bbb C}P^n$ defined by $(A,[v]) \mapsto [Av]$ is transitive. Show that a matrix $A \in GL(n+1,{\Bbb C})$ fixes all points of ${\Bbb C}P^n$ if and only if it is a multiple of the indentity.

Definition 2.1   The projective group $PGL(n,{\Bbb C}), n \geq 2$, is the quotient of $GL(n,{\Bbb C})$ by the equivalence relation: two matrices are equivalent if they are multiple of each other.

Exercise 2.2 (00)   Use exercise 2.1 to define a transitive action of $PGL(n+1,{\Bbb C})$ on ${\Bbb C}P^n$. Show that if an element of the projective group fixes all points in projective space, then it is the identity.

Exercise 2.3   Identify ${\Bbb C}P^1$ with ${\Bbb C}$ plus the point at infinity and let

$$A := \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

be an invertible (complex) matrix. Show that the effect of the action of $A$ on a complex number $\zeta$ equals $(a\zeta + b)/(c\zeta + d)$.

Transformations of the form $\zeta \mapsto (a\zeta + b)/(c\zeta + d)$ are commonly called Moebius transformations or linear fractional transformations.

Exercise 2.4 (00)   Using the formulas of chapter 2, it is easy to extend the first and second fundamental theorems of projective geometry to the complex setting.

1. Show that there exists a unique transformation of the form
   $$\zeta \longmapsto \frac{a\zeta + b}{c\zeta + d}$$
   that takes a triple of distinct complex numbers $z_{0}, z_{1}, z_{2}$ to $\infty, 0, 1$.
2. Write the formula for the projective transformation that takes $\infty, 0, 1$ to $z_{0}, z_{1}, z_{2}$ (i.e., the inverse of the above transformation).
3. Write the formula for the transformation that takes $z_{0}, z_{1}, z_{2}$ to $z'_{0},z'_{1},z'_{2}$.

As a trivial consequence of this exercise we have:

Theorem 2.1 (First fundamental theorem)   If $x,y,z$ and $x',y',z'$ are two triples of distinct points on the complex projective line, there exits a unique projective transformation that sends $x$ to $x'$, $y$ to $y'$ and $z$ to $z'$.

Definition 2.2   Let $p_{0},\dots,p_{3}$ be a quadruple of points on the complex projective line with $p_{0},p_{1}$, and $p_{2}$ different from each other, and let $T$ be the unique projective transformation taking $p_{0},p_{1}$, and $p_{2}$ respectively onto $\infty, 0$, and $1$. The cross-ratio $[p_{0},p_{1},p_{2},p_{3}]$ is defined as the point $T(p_{3}) \in {\Bbb C}\cup \{\infty\}$.

Exercise 2.5 (00)   By indentifying the complex projective line with ${\Bbb C}\cup \{\infty\}$, verify that the cross-ratio of four complex numbers $z_{0},\dots,z_{3}$ can be written as:

$$[z_{0},z_{1},z_{2},z_{3}] = \frac{z_{2} - z_{0}}{z_{2} - z_{1}} \frac{z_{3} - z_{1}}{z_{3} - z_{0}}$$

Exercise 2.6 (00)   Show that if $T$ is a projective transformation and $p_{0},\dots,p_{3}$ is a quadruple of points on the projective line with $p_{0},p_{1}$, and $p_{2}$ different from each other, then the cross-ratio $[p_{0},p_{1},p_{2},p_{3}]$ equals $[T(p_{0}), T(p_{1}), T(p_{2}),T(p_{3})]$.

Theorem 2.2 (Second fundamental theorem)   A map from the complex projective line to itself is a projective transformation if and only if it preserves cross-ratios.

Exercise 2.7 (00)   Prove theorem 2.2.

The preceding exercises -- exact copies of those of chapter 2 -- may lull the reader into believing that the geometry of the complex projective line is similar to that of the real projective line. We will presently see that the former is infinitely richer and more beautiful. For one thing, circles play a privileged role in complex projective geometry.

Remark. From now on straight lines and circles will be simply called circles. The intuitive idea is that a straight line is a circle of infinite radius, or just the stereographic projection of a circle passing through the north pole.

Exercise 2.8 (10)   Show that four points in ${\Bbb C}\cup \{\infty\}$ are on the same circle if and only if their cross-ratio is a real number.

Exercise 2.9 (05)   Write the equation of the circle passing through points $(0,0), (0,2)$ and $(1,1)$.

An obvious consequence of exercise 2.8 is the following important result.

Proposition 2.1   Moebius transformations send circles to circles.

Exercise 2.10 (10)   Show that given any two circles there is always a Moebius transformation that takes one circle to the other.

To better understand Moebius transformations, let us consider three particular cases. The first two cases are already familiar: transformations of the form $z \mapsto z + b$ are translations of the number $z$ thought of as a vector on the plane, transformations of the form $z \mapsto az, a \neq 0$ are dilations combined with rotations around the origin. The third case, the inversion $z \mapsto 1/z$, is more interesting.

Exercise 2.11 (15)   Take a point $(u,v,0) = u + iv$ on ${\Bbb C}= {\Bbb R}^2 \subset {\Bbb R}^3$ and let $(x,y,z)$ be the inverse image of $u + iv$ by stereographic projection from the north pole of the unit sphere. Trace a line from $(x,y,z)$ to the south pole and mark the point where this line intersects ${\Bbb R}^2$. Show that this point is $1/(u+iv)$.

The reader should always keep in mind that Moebius transformations are just complex projective transformations when seen in the model that identifies ${\Bbb C}P^1$ with ${\Bbb C}$ plus a point at infinity. Changing from this model to the sphere usually leads to new insights such as the one in the exercise above. Another example is supplied by the following exercise:

Exercise 2.12 (*15)   Show that a Moebius transformation $z \mapsto (az + b)/(cz + d)$ induces a rotation on the sphere if and only if $d = \bar{a}$ and $c = -\bar{b}$. In other words if and only if the matrix

$$A := \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

is unitary. Hint: rotations fix a pair of antipodal points.

Since we can multiply all entries of $A$ by the same nonzero complex number without changing the transformation, we may assume that the determinant of $A$ equals 1. Use this to identify the group $SU(2)$ as the double cover of the group of rotations $SO(3)$.

Exercise 2.13 (20)   Show that any Moebius transformation is the composition of translations, dilations, rotations, and inversions.

Exercise 2.14 (10)   The previous result allows us to give a new proof that circles are preserved under Moebius transformations. In fact, it suffices to prove that the inversion $z \mapsto 1/z$ sends circles to circles.

• Let $A$ and $C$ be real numbers and let $B$ be a complex number. Show that the equation $Az\bar{z} + Bz + \bar{B}\bar{z} + C = 0$ defines a circle if and only if $AC - \vert B\vert^2 < 0$. Conversely, show that any circle has an equation of this form.
• Use this to show that Moebius transformations send circles to circles.
• An anti-homography is a transformation of the form $z \mapsto (a\bar{z} + b)/(c\bar{z} + b)$, with $ad-bc \neq 0$. Show that anti-homographies also send circles to circles.

Besides sending circles to circles, Moebius transformations have the important property that they preserve angles and orientation. To place this in the proper context, we will study all transformations that satisfy this property.

Exercise 2.15 (20)   Show that the linear transformation from ${\Bbb R}^2$ to itself defined by the invertible matrix

$$A := \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

preserves angles and orientation if and only if $a = d$ and $c = -b$. In other words, $A$ represents multiplication by a nonzero complex number. Hint: $A$ preserves orientation if and only if its determinant is positive.

Definition 2.3   A smooth map $f$ from an open subset of ${\Bbb R}^2$ to ${\Bbb R}^2$ is said to preserve angles if at each point $(x,y)$ in its domain the differential

$$D_{(x,y)}f : {\Bbb R}^2 \longrightarrow {\Bbb R}^2$$

is a linear transformation that preserves angles. Likewise, $f$ is said to preserve orientation if its differential is a linear map that preserves the orientation.

A smooth map from an open subset of ${\Bbb R}^2$ to ${\Bbb R}^2$ that preserves angles and orientations is said to be conformal.

Exercise 2.16 (10)   This exercise shows that conformal maps are characterized by a simple system of differential equations.

1. Show that a smooth map $f(x,y) := (u(x,y),v(x,y))$ from an open subset of ${\Bbb R}^2$ to ${\Bbb R}^2$ is conformal if and only if the Cauchy-Riemann equations hold:
   $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$    and $$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} .$$

2. Let $\frac{\partial}{\partial \bar{z}} := {1 \over 2}(\frac{\partial}{\partial x} +i \frac{\partial}{\partial y})$. Show that $f = u + iv$ is conformal if and only if $\frac{\partial f}{\partial \bar{z}}$ is identically zero.
3. Show that if $f = (u,v)$ is a conformal transformation, then $u$ and $v$ are harmonic functions:
   $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial ... ... = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0 .$$

Exercise 2.17 (10)   Let $f$ and $g$ be two conformal maps defined on some open subset of ${\Bbb R}^2$. In this exercise, we will consider the values of $f$ and $g$ to be complex numbers rather than just points or vectors in ${\Bbb R}^2$.

1. Show that the function $f + g$ is a conformal map.
2. Show that the function $f \cdot g$ (complex multiplication) is a conformal map. Conclude that any complex polynomial $P(z) = a_{0} + a_{1}z + \cdots a_{n}z^n$ defines a conformal map.
3. Assuming that $f$ is not identically zero, show that the function $1/f$ is conformal on its domain of definition. Conclude that a rational function (i.e., a quotient of two polynomials) is conformal on its domain of definition.

The previous exercise proves the following important result:

Theorem 2.3   Moebius transformations are conformal.

Using that Moebius transformations preserve angles, we can give a simple proof that the stereographic projection preserves angles.

Exercise 2.18   Use exercise 1.7, the above theorem, and the fact that rotations on the sphere can be represented by Moebius transformations to show that stereographic projection preserves angles.