is the unit disc on the complex plane
is the upper half plane on the complex plane,
is the band between and .
Definition 1. Hyperbolic metric on the unit disc.
The hyperbolic metric on the unit disc is defined as
If is a conformal mapping, where , then we can also define the hyperbolic metric on the domain U,
From above and is a conformal mapping which maps the upper half plane onto the unit disc From above formula, we can calculate the hyperbolic metric on is
The hyperbolic metric on the band is
Similarly, we can define the one dimensional hyperbolic metric. On the real line , if the interval , then the restriction of the hyperbolic metric on the unit disc is
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 . Since there exists a linear map which maps a to -1 and b to 1, i.e. . Its derivative is . Therefore, the hyperbolic metric on the interval I is
Moreover, assume , then the hyperbolic distance between c and d is
If we use the notation of cross ratio, then assume . Therefore, the hyperbolic distance between c and d in the interval (a,b) equals to
Definition 2. (Curvature of conformal metric)
Let be a positive function on an open subset . Then the curvature of the metric is given by
where is the Laplacian operator
Remark. Use the identities
The curvature of hyperbolic metric of the unit disc , the upper half plane and the band is -1.
If is a conformal mapping, where is an open subset of the complex plane . From above, the hyperbolic metric on U is
Then the curvature of the metric is .
On the complex sphere , the sphere metric on is defined as
Then the curvature of the sphere metric is 1.