Plane Hyperbolic Geometry

Assume

\mathbb{D}=\{ z: |z|<1\} is the unit disc on the complex plane \mathbb{C},

\mathbb{H}=\{z: \Im{z}>0\} is the upper half plane on the complex plane,

\mathbb{B}=\{ z: |\Im{z}|<\pi/2\} is the band between y=-\pi/2 and y=\pi/2.

Definition 1. Hyperbolic metric on the unit disc.

The hyperbolic metric on the unit disc \mathbb{D} is defined as

\rho_{\mathbb{D}}(z)=\frac{2}{1-|z|^{2}} |dz| for all z \in \mathbb{D} .

If \phi : U \rightarrow \mathbb{D} is a conformal mapping, where U \subseteq \mathbb{C}, then we can also define the hyperbolic metric on the domain U,

\rho_{U}(z)=\frac{2 |\phi^{'}(z)|}{1-|\phi(z)|^{2}} |dz| for all z\in U.

From above and \phi(z)=(z-i)/(z+i) is a conformal mapping which maps the upper half plane \mathbb{H} onto the unit disc \mathbb{D}. From above formula, we can calculate the hyperbolic metric on \mathbb{H} is

\rho_{\mathbb{H}}(z)=\frac{1}{\Im{z}} |dz| for all z\in \mathbb{H}.

The hyperbolic metric on the band \mathbb{B} is

\rho_{\mathbb{B}}(z)=\frac{1}{\cos \Im{z} } |dz| for all z\in \mathbb{B}.

Similarly, we can define the one dimensional hyperbolic metric. On the real line \mathbb{R}, if the interval I=(-1,1), then the restriction of the hyperbolic metric on the unit disc \mathbb{D} is

\rho_{I}(x)= \frac{2}{1-x^{2}} dx for all x \in (-1,1).

This is called the hyperbolic metric of the interval I.

Using the same idea, we can extend the definition of hyperbolic metric on any real interval I=(a,b). Since there exists a linear map \phi which maps a to -1 and b to 1, i.e. \phi(x)=(2x-b-a)/(b-a). Its derivative is \phi^{'}(x)= 2/(b-a). Therefore, the hyperbolic metric on the interval I is

\rho_{(a,b)}(x)=\frac{2|\phi^{'}(x)|}{1-|\phi(x)|^{2}} dx= \frac{b-a}{(x-a)(b-x)} dx= (\frac{1}{x-a}+ \frac{1}{b-x}) dx for all x\in (a,b).

Moreover, assume (c,d) \subseteq (a,b), then the hyperbolic distance between c and d is

\int_{c}^{d} \rho_{(a,b)}(x) dx = \int_{c}^{d} (\frac{1}{x-a} + \frac{1}{b-x}) dx = (\ln\frac{x-a}{b-x}) |_{x=c}^{x=d} = \ln \frac{(d-a)(b-c)}{(b-d)(c-a)}.

If we use the notation of cross ratio, then assume l=(a,c), j=(c,d), r=(d,b), t=(a,b). Therefore, the hyperbolic distance between c and d in the interval (a,b) equals to

\ln \frac{(|l|+|j|)\cdot (|j|+|r|)}{|l| \cdot |r|} = \ln (1+ \frac{|t|\cdot |j|}{|l| \cdot |r|}) = \ln (1+ Cr(t,j)),

where Cr(t,j)= (|t|\cdot |j|) / (|l| \cdot |r|).

Definition 2. (Curvature of conformal metric)

Let \rho be a C^{2} positive function on an open subset U \subseteq \mathbb{C}. Then the curvature of the metric \rho(z)|dz| is given by

K(z)=-\frac{(\Delta \ln \rho)(z)}{\rho^{2}(z)},

where \Delta is the Laplacian operator \Delta= \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}.

Remark. Use the identities

\frac{\partial}{\partial \overline{z}} =\frac{1}{2} (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}),

\frac{\partial}{\partial z} =\frac{1}{2} (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}),

we get

\Delta=\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} = 4 \frac{\partial^{2}}{\partial z \partial \overline{z}}.

Theorem 1. 

The curvature of hyperbolic metric of the unit disc \mathbb{D}, the upper half plane \mathbb{H} and the band \mathbb{B} is -1.

Theorem 2.

If \phi: U\rightarrow \mathbb{D} is a conformal mapping, where U \subseteq \mathbb{C} is an open subset of the complex plane \mathbb{C}. From above, the hyperbolic metric on U is

\rho_{U}(z)=\frac{2 |\phi^{'}(z)|}{1-|\phi(z)|^{2}} |dz| for all z \in U\subseteq \mathbb{C}.

Then the curvature of the metric \rho_{U}(z) is -1.

Theorem 3.

On the complex sphere \hat{\mathbb{C}}, the sphere metric on \hat{\mathbb{C}} is defined as

\rho(z)=\frac{1}{1+|z|^{2}} |dz| for all z\in \hat{\mathbb{C}}.

Then the curvature of the sphere metric is 1.

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