Hyperbolic metric
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 is defined by the line element
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where is the line element of Euclidean length. The introduction of the hyperbolic metric in
leads to a model of Lobachevskii geometry. In this model the role of straight lines is played by Euclidean circles orthogonal to
and lying in
; the circle
plays the role of the improper point. Fractional-linear transformations of
onto itself serve as the motions in it. The hyperbolic length of a curve
lying inside
is defined by the formula
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The hyperbolic distance between two points and
of
is
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The set of points of whose hyperbolic distance from
,
, does not exceed a given number
,
, i.e. the hyperbolic disc in
with hyperbolic centre at
and hyperbolic radius
, is a Euclidean disc with centre other than
if
.
The hyperbolic area of a domain lying in
is defined by the formula
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The quantities ,
and
are invariant with respect to fractional-linear transformations of
onto itself.
The hyperbolic metric in any domain of the
-plane with at least three boundary points is defined as the pre-image of the hyperbolic metric in
under the conformal mapping
of
onto
; its line element is defined by the formula
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A domain with at most two boundary points can no longer be conformally mapped onto a disc. The quantity
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is called the density of the hyperbolic metric of . The hyperbolic metric of a domain
does not depend on the selection of the mapping function or of its branch, and is completely determined by
. The hyperbolic length of a curve
located in
is found by the formula
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The hyperbolic distance between two points and
in a domain
is
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where is any function conformally mapping
onto
. A hyperbolic circle in
is, as in the case of the disc
, a set of points in
whose hyperbolic distance from a given point of
(the hyperbolic centre) does not exceed a given positive number (the hyperbolic radius). If the domain
is multiply connected, a hyperbolic circle in
is usually a multiply-connected domain. The hyperbolic area of a domain
lying in
is found by the formula
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The quantities ,
and
are invariant under conformal mappings of
(one of the main properties of the hyperbolic metric in
).
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) |
Comments
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 be a domain,
and
. Denote by
the set of holomorphic mappings
,
the unit ball in
. Then the (infinitesimal version of the) Carathéodory metric is
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and the (infinitesimal version of the) Kobayashi distance is
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Instead of 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) |
Hyperbolic metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_metric&oldid=17817