8.2 Conformal Metrics and Curvature

Let be a region and let be a positive function. The conformal metric (in the following chapters when we refer to metric we mean conformal) induced by is given by

The term “conformality” is explained in the previous section (note that this specific usage has little to do with holomorphy). The distance between two points is defined as

where the infimum is taken over all piecewise smooth curves in joining and .

A metric is said to be regular. The (Gaussian) curvature of the regular metric at is defined as

where is the Laplacian operator. This is the same definition as the Gaussian curvature in (8.2.6).

The three following metrics are of particular interest in complex differential geometry:

• Perhaps the most trivial metric is the Euclidean metric (also known as the parabolic metric) on , and is given by

  The Euclidean distance between two points is

  is the length of the straight line segment connecting and . The group formed by all transformations in the form of (where and ) is known as the group of rigid motions, or more abstractly, the special Euclidean group of order , denoted by , intuitively consists of all rotations and translations and their compositions, while the Euclidean group consists of reflections in the form of . Obviously, the Euclidean metric is invariant under both groups.

  From (8.2.1), we find that Euclidean metric has curvature .

• The Poincaré metric (also referred to as the hyperbolic metric) on is given by

  In Lemma 3.5.2, it was shown that the metric is invariant under .

  We will now calculate the Poincaré distance between two points . First assume the case where and . Consider a piecewise smooth curve parameterized by connecting and ; or in other words

  where and are real-valued functions. Then

  Assuming that is in the form of where , we have

  Hence, the Poincaré distance between and is given by

  and the straight line segment connecting the two points is a geodesic (path of least length under a metric or other criteria). For fixed since , by the Schwarz–Pick Lemma (Lemma 3.5.2), we have

  by the invariance under . Now let and be arbitrary points in . The Möbius transformation

  maps to and maps to . Hence, we have

  which is the Poincaré distance (or hyperbolic distance) between and . The infimum is attained along the geodesic curve parameterized by

  for . By Theorem 5.1.4, the geodesic is either an arc or a straight line segment passing through and . Since is orthogonal to the straight line passing through and , by the conformality of , is orthogonal to the circular (or straight line) extension of the geodesic curve.

  As a consequence of the Schwarz–Pick Lemma (Lemma 3.5.2), for any is holomorphic, we have

  where equality is attained iff . The Poincaré metric has constant negative curvature since

  where and .

• The spherical metric (also referred to as the elliptic metric) on is given by

  Under the inverse stereographic projection of , for a given , the corresponding point in is

  If we let and be two points in , the distance between the two points is the length of the shortest arc (a subset of great circle passing the two points). By considering and as vectors from , this distance is equal to

  Notice that the fraction within the square root is a product between a complex number and its conjugate. Thus, this distance is equal to

  in the extended complex plane. Let . It follows that

  where we have taken the liberty to coalesce orders for simplification. Since

  the metric as defined in (8.2.3) has a clear geometric meaning: the distance between two points and under the metric in (8.2.3) is the shortest distance between the corresponding points in , or their spherical distance.

  Thus, if curve joins and , we have

  which attains its infimum when the inverse stereographic projection of is a great circle of . Thus, is known as the spherical metric.

  The corresponding curvature is given by

  where and . This can also be verified by computing the principal curvatures of the unit sphere, which are both one.

The importance of the selected regions lies in the uniformization to be mentioned in @ sec:riemannsurfaces.

Let and be two open regions in such that is univalent (implying that by Lemma 5.1.1). If is a metric on , then

defines a metric on , referred to as the metric pullback of by .

Curvature as defined in (8.2.1) is invariant under pullbacks of conformal mappings, or in the case above, we now aim to show that (under assumptions of regularity)

By explicit definition,

Since , is harmonic on with a vanishing Laplacian. Hence,

For a given metric , if there is some other parameterization such that , is conformal, then the relation is given by . Under differing parameterizations of a metric , we once again have the invariance of curvature.