# Hyperbolic metric

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hyperbolic measure

A metric in a domain of the complex plane with at least three boundary points that is invariant under automorphisms of this domain.

The hyperbolic metric in the disc $E = \{ {z } : {| z | < 1 } \}$ is defined by the line element

$$d \sigma _ {z} = \ \frac{| dz | }{1 - | z | ^ {2} } ,$$

where $| dz |$ is the line element of Euclidean length. The introduction of the hyperbolic metric in $E$ leads to a model of Lobachevskii geometry. In this model the role of straight lines is played by Euclidean circles orthogonal to $| z | = 1$ and lying in $E$; the circle $| z | = 1$ plays the role of the improper point. Fractional-linear transformations of $E$ onto itself serve as the motions in it. The hyperbolic length of a curve $L$ lying inside $E$ is defined by the formula

$$\mu _ {E} ( L) = \ \int\limits _ { L } \frac{| dz | }{1 - | z | ^ {2} } .$$

The hyperbolic distance between two points $z _ {1}$ and $z _ {2}$ of $E$ is

$$r _ {E} ( z _ {1} , z _ {2} ) = \ { \frac{1}{2} } \mathop{\rm ln} \ \frac{| 1 - z _ {1} \overline{ {z _ {2} }}\; | + | z _ {1} - z _ {2} | }{| 1 - z _ {1} \overline{ {z _ {2} }}\; | - | z _ {1} - z _ {2} | } .$$

The set of points of $E$ whose hyperbolic distance from $z _ {0}$, $z _ {0} \in E$, does not exceed a given number $R$, $R > 0$, i.e. the hyperbolic disc in $E$ with hyperbolic centre at $z _ {0}$ and hyperbolic radius $R$, is a Euclidean disc with centre other than $z _ {0}$ if $z _ {0} \neq 0$.

The hyperbolic area of a domain $B$ lying in $E$ is defined by the formula

$$\Delta _ {E} ( B) = \ {\int\limits \int\limits } _ { B } \frac{dx dy }{( 1 - | z | ^ {2} ) ^ {2} } ,\ \ z = x + iy.$$

The quantities $\mu _ {E} ( L)$, $r _ {E} ( z _ {1} , z _ {2} )$ and $\Delta _ {E} ( B)$ are invariant with respect to fractional-linear transformations of $E$ onto itself.

The hyperbolic metric in any domain $D$ of the $z$- plane with at least three boundary points is defined as the pre-image of the hyperbolic metric in $E$ under the conformal mapping $\zeta = \zeta ( z)$ of $D$ onto $E$; its line element is defined by the formula

$$d \sigma _ {z} = \ \frac{| \zeta ^ \prime ( z) | | dz | }{1 - | \zeta ( z) | ^ {2} } .$$

A domain with at most two boundary points can no longer be conformally mapped onto a disc. The quantity

$$\rho _ {D} ( z) = \ \frac{| \zeta ^ \prime ( z) | }{1 - | \zeta ( z) | ^ {2} }$$

is called the density of the hyperbolic metric of $D$. The hyperbolic metric of a domain $D$ does not depend on the selection of the mapping function or of its branch, and is completely determined by $D$. The hyperbolic length of a curve $L$ located in $D$ is found by the formula

$$\mu _ {D} ( L) = \ \int\limits _ { L } \rho _ {D} ( z) | dz | .$$

The hyperbolic distance between two points $z _ {1}$ and $z _ {2}$ in a domain $D$ is

$$r _ {D} ( z _ {1} , z _ {2} ) = { \frac{1}{2} } \mathop{\rm ln} \ \frac{| 1 - \zeta ( z _ {1} ) \overline{ {\zeta ( z _ {2} ) }}\; | + | \zeta ( z _ {1} ) - \zeta ( z _ {2} ) | }{| 1 - \zeta ( z _ {1} ) \overline{ {\zeta ( z _ {2} ) }}\; | - | \zeta ( z _ {1} ) - \zeta ( z _ {2} ) | } ,$$

where $\zeta ( z)$ is any function conformally mapping $D$ onto $E$. A hyperbolic circle in $D$ is, as in the case of the disc $E$, a set of points in $D$ whose hyperbolic distance from a given point of $D$( the hyperbolic centre) does not exceed a given positive number (the hyperbolic radius). If the domain $D$ is multiply connected, a hyperbolic circle in $D$ is usually a multiply-connected domain. The hyperbolic area of a domain $B$ lying in $D$ is found by the formula

$$\Delta _ {D} ( B) = \ {\int\limits \int\limits } _ { B } \rho _ {D} ^ {2} ( z) dx dy.$$

The quantities $\mu _ {D} ( L)$, $r _ {D} ( z _ {1} , z _ {2} )$ and $\Delta _ {D} ( B)$ are invariant under conformal mappings of $D$( one of the main properties of the hyperbolic metric in $D$).

#### References

 [1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) [2] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)

Generalizations to higher-dimensional domains (mainly strongly pseudo-convex domains) are, e.g., the Carathéodory metric, the Kobayashi metric and the Bergman metric (for the latter see Bergman kernel function).

Let $\Omega \subset \mathbf C ^ {n}$ be a domain, $z \in \Omega$ and $\xi \in \mathbf C ^ {n}$. Denote by $B ( \Omega )$ the set of holomorphic mappings $f : \Omega \rightarrow B$, $B$ the unit ball in $\mathbf C ^ {n}$. Then the (infinitesimal version of the) Carathéodory metric is

$$F _ {C} ( z, \xi ) = \ \sup _ {\begin{array}{c} f \in B ( \Omega ) \\ f( z) = 0 \end{array} } \ \left | \sum _ { j= } 1 ^ { n } \frac{\partial f }{\partial z _ {j} } ( z) \cdot \xi _ {j} \right | ,$$

and the (infinitesimal version of the) Kobayashi distance is

$$F _ {K} ( z, \xi ) =$$

$$= \ \inf \{ \alpha : \alpha > 0 \textrm{ and there is a holomorphic mapping } \$$

$${} f : B \rightarrow \Omega \textrm{ with } f( z)= 0, ( f ^ { \prime } ( 0) ) ( 1 , 0 \dots 0) = \xi / \alpha \} .$$

Instead of $B$ sometimes other domains (e.g. the unit polydisc) are taken. (See [a2], [a3].)

One correspondingly defines for these metrics distance and area.

#### References

 [a1] L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973) [a2] S. Lang, "Introduction to complex hyperbolic spaces" , Springer (1987) [a3] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
How to Cite This Entry:
Hyperbolic metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_metric&oldid=47287
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article