Math 411: Honours Complex Variables - University of Alberta

112 CHAPTER 17. THE RIEMANN MAPPING THEOREM
Proposition 17.1. Let D1,D2 ⊂ C be open and connected such that D1 is simply
connected, and suppose that D1 and D2 are biholomorphically equivalent. Then D2 is
simply connected.
Proof. Let f: D1 → D2 be biholomorphic. Let g: D2 → C be holomorphic, and let γ
be a closed curve in D2. Since f−1 ◦γ is a closed curve in the simply connected set
D1, we see that
� � �
g(ζ)dζ = g(ζ)dζ = g(f(ζ))f ′ (ζ)dζ = 0.
γ
f◦f −1 ◦γ
f −1 ◦γ
Example. Let r,R ∈ [0,∞]besuch that r < R. Then Ar,R(0)isnot biholomorphically
equivalent to D or C.
Biholomorphic maps have a very interesting geometric property.
Given and open set D ⊂ R N and curves γ1,γ2: [0,1] → D, suppose there exist
t1,t2 ∈ (0,1) such that γ1(t1) = γ2(t2) = x0. In order to define the angle between γ1
and γ2 at x0, we further suppose that there exists ǫ > 0 such that γj is differentiable
on (tj − ǫ,tj + ǫ) for j = 1,2 with γ ′ j (tj) �= 0. The angle is then defined to be the
unique θ ∈ [0,π] such that
cosθ = γ′ 1 (t1)·γ ′ 2 (t2)
|γ ′ 1(t1)||γ ′ 2(t2)| .
Given two open sets D1,D2 ⊂ R N , a differentiable map f : D1 → D2 is called angle
preserving at x0 ∈ D1 if, for any two curves γ1 and γ2 in D1, the angle between f ◦γ1
and f ◦γ2 at f(x0) is the same between γ1 and γ2 at x0.
Recall that a real N × N matrix A is called orthogonal if it is invertible with
A −1 = A t .
Lemma 17.1. Let D1,D2 ⊂ R N be open, let x0 ∈ D1, and let f : D1 → D2 be
differentiable such that Jf(x0) is orthogonal. Then f is angle preserving at x0.
Proof. Let γ1 and γ2 be two curves in D1 satisfying the necessary requirements, and
note that
cosine of the angle between f ◦γ1 and f ◦γ2 at f(x0)
= (f ◦γ1) ′ (t1)·(f ◦γ2) ′ (t2)
|(f ◦γ1) ′ (t1)||(f ◦γ2) ′ (t2)|
= Jf(x0)γ ′ 1(t1)·Jf(x0)γ ′ 2(t2)
|Jf(x0)γ ′ 1(t1)||Jf(x0)γ ′ 2(t2)| , by the chain rule,
= Jf(x0) t Jf(x0)γ ′ 1(t1)·γ ′ 2(t2)
|Jf(x0)γ ′ 1 (t1)||Jf(x0)γ ′ 2 (t2)|
= γ′ 1 (t1)·γ ′ 2 (t2)
|γ ′ 1(t1)||γ ′ 2(t2)|
= cosine of the angle between γ1 and γ2 at x0.