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''on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248401.png" />''
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A rule that associates with each local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248402.png" />, mapping a parameter neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248403.png" /> into the closed complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248404.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248405.png" />), a real-valued function
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248406.png" /></td> </tr></table>
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''on a Riemann surface  $  R $''
  
such that for all local parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248408.png" /> for which the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c0248409.png" /> is not empty, the following relation holds:
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A rule that associates with each local parameter  $  z $,
 +
mapping a parameter neighbourhood  $  U \subset  R $
 +
into the closed complex plane  $  \overline{\mathbf C}\; $(
 +
$  z : U \rightarrow \overline{\mathbf C}\; $),  
 +
a real-valued function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484010.png" /></td> </tr></table>
+
$$
 +
\rho _ {z} : z ( U)  \rightarrow \
 +
[ 0 , + \infty ]
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484011.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484013.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484014.png" />. A conformally-invariant metric is often denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484015.png" />, to which the indicated invariance with respect to the choice of the local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484016.png" /> is attributed.
+
such that for all local parameters  $  z _ {1} :  U \rightarrow \overline{\mathbf C}\; $
 +
and  $  z _ {2} : U _ {2} \rightarrow \overline{\mathbf C}\; $
 +
for which the intersection  $  U _ {1} \cap U _ {2} $
 +
is not empty, the following relation holds:
  
Every linear differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484017.png" /> (or [[Quadratic differential|quadratic differential]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484018.png" />) induces a conformally-invariant metric, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484019.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484020.png" />). The notion of a conformally-invariant metric, being a very general form of defining conformal invariants, enables one to introduce that of the length of curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024840/c02484021.png" /> as well as the notion of the extremal length and the modulus of families of curves (see [[Extremal metric, method of the|Extremal metric, method of the]], and also [[#References|[1]]]). The definition of a conformally-invariant metric can be carried over to Riemann varieties of arbitrary dimension.
+
$$
 +
 
 +
\frac{\rho _ {z _ {2}  } ( z _ {2} ( p) ) }{\rho _ {z _ {1}  } ( z _ {1} ( p) ) }
 +
  = \
 +
\left |
 +
 
 +
\frac{d z _ {1} ( p) }{d z _ {2} ( p) }
 +
\
 +
\right | \ \
 +
( \forall p \in U _ {1} \cap U _ {2} ) ,
 +
$$
 +
 
 +
where  $  z ( U) $
 +
is the image of  $  U $
 +
in  $  \overline{\mathbf C}\; $
 +
under  $  z $.
 +
A conformally-invariant metric is often denoted by the symbol  $  \rho ( z)  | d z | $,
 +
to which the indicated invariance with respect to the choice of the local parameter  $  z $
 +
is attributed.
 +
 
 +
Every linear differential $  \lambda ( z)  d z $(
 +
or [[Quadratic differential|quadratic differential]] $  Q ( z)  d z  ^ {2} $)  
 +
induces a conformally-invariant metric, $  | \lambda ( z) | \cdot | d z | $(
 +
or $  | Q ( z)  |  ^ {1/2} |  d z | $).  
 +
The notion of a conformally-invariant metric, being a very general form of defining conformal invariants, enables one to introduce that of the length of curves on $  R $
 +
as well as the notion of the extremal length and the modulus of families of curves (see [[Extremal metric, method of the|Extremal metric, method of the]], and also [[#References|[1]]]). The definition of a conformally-invariant metric can be carried over to Riemann varieties of arbitrary dimension.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.J. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  L.V. Ahlfors,  "The complex analytic structure of the space of closed Riemann surfaces"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 45–66</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  L.V. Ahlfors,  "On quasiconformal mappings"  ''J. d'Anal. Math.'' , '''3'''  (1954)  pp. 1–58</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  L.V. Ahlfors,  "Correction to  "On quasiconformal mappings" "  ''J. d'Anal. Math.'' , '''3'''  (1954)  pp. 207–208</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top">  L. Bers,  "Quasi-conformal mappings and Teichmüller's theorem"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 89–119</TD></TR><TR><TD valign="top">[3e]</TD> <TD valign="top">  L. Bers,  "Spaces of Riemann surfaces"  J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press  (1960)  pp. 349–361</TD></TR><TR><TD valign="top">[3f]</TD> <TD valign="top">  L. Bers,  "Simultaneous uniformization"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 94–97</TD></TR><TR><TD valign="top">[3g]</TD> <TD valign="top">  L. Bers,  "Holomorphic differentials as functions of moduli"  ''Bull. Amer. Math. Soc.'' , '''67'''  (1961)  pp. 206–210</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.J. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  L.V. Ahlfors,  "The complex analytic structure of the space of closed Riemann surfaces"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 45–66</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  L.V. Ahlfors,  "On quasiconformal mappings"  ''J. d'Anal. Math.'' , '''3'''  (1954)  pp. 1–58</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top">  L.V. Ahlfors,  "Correction to  "On quasiconformal mappings" "  ''J. d'Anal. Math.'' , '''3'''  (1954)  pp. 207–208</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top">  L. Bers,  "Quasi-conformal mappings and Teichmüller's theorem"  R. Nevanlinna (ed.)  et al. (ed.) , ''Analytic functions'' , Princeton Univ. Press  (1960)  pp. 89–119</TD></TR><TR><TD valign="top">[3e]</TD> <TD valign="top">  L. Bers,  "Spaces of Riemann surfaces"  J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press  (1960)  pp. 349–361</TD></TR><TR><TD valign="top">[3f]</TD> <TD valign="top">  L. Bers,  "Simultaneous uniformization"  ''Bull. Amer. Math. Soc.'' , '''66'''  (1960)  pp. 94–97</TD></TR><TR><TD valign="top">[3g]</TD> <TD valign="top">  L. Bers,  "Holomorphic differentials as functions of moduli"  ''Bull. Amer. Math. Soc.'' , '''67'''  (1961)  pp. 206–210</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Lectures on quasiconformal mappings" , v. Nostrand  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Lectures on quasiconformal mappings" , v. Nostrand  (1966)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


on a Riemann surface $ R $

A rule that associates with each local parameter $ z $, mapping a parameter neighbourhood $ U \subset R $ into the closed complex plane $ \overline{\mathbf C}\; $( $ z : U \rightarrow \overline{\mathbf C}\; $), a real-valued function

$$ \rho _ {z} : z ( U) \rightarrow \ [ 0 , + \infty ] $$

such that for all local parameters $ z _ {1} : U \rightarrow \overline{\mathbf C}\; $ and $ z _ {2} : U _ {2} \rightarrow \overline{\mathbf C}\; $ for which the intersection $ U _ {1} \cap U _ {2} $ is not empty, the following relation holds:

$$ \frac{\rho _ {z _ {2} } ( z _ {2} ( p) ) }{\rho _ {z _ {1} } ( z _ {1} ( p) ) } = \ \left | \frac{d z _ {1} ( p) }{d z _ {2} ( p) } \ \right | \ \ ( \forall p \in U _ {1} \cap U _ {2} ) , $$

where $ z ( U) $ is the image of $ U $ in $ \overline{\mathbf C}\; $ under $ z $. A conformally-invariant metric is often denoted by the symbol $ \rho ( z) | d z | $, to which the indicated invariance with respect to the choice of the local parameter $ z $ is attributed.

Every linear differential $ \lambda ( z) d z $( or quadratic differential $ Q ( z) d z ^ {2} $) induces a conformally-invariant metric, $ | \lambda ( z) | \cdot | d z | $( or $ | Q ( z) | ^ {1/2} | d z | $). The notion of a conformally-invariant metric, being a very general form of defining conformal invariants, enables one to introduce that of the length of curves on $ R $ as well as the notion of the extremal length and the modulus of families of curves (see Extremal metric, method of the, and also [1]). The definition of a conformally-invariant metric can be carried over to Riemann varieties of arbitrary dimension.

References

[1] J.J. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[2] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)
[3a] L.V. Ahlfors, "The complex analytic structure of the space of closed Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 45–66
[3b] L.V. Ahlfors, "On quasiconformal mappings" J. d'Anal. Math. , 3 (1954) pp. 1–58
[3c] L.V. Ahlfors, "Correction to "On quasiconformal mappings" " J. d'Anal. Math. , 3 (1954) pp. 207–208
[3d] L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 89–119
[3e] L. Bers, "Spaces of Riemann surfaces" J.A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 349–361
[3f] L. Bers, "Simultaneous uniformization" Bull. Amer. Math. Soc. , 66 (1960) pp. 94–97
[3g] L. Bers, "Holomorphic differentials as functions of moduli" Bull. Amer. Math. Soc. , 67 (1961) pp. 206–210

Comments

References

[a1] L.V. Ahlfors, "Lectures on quasiconformal mappings" , v. Nostrand (1966)
How to Cite This Entry:
Conformally-invariant metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformally-invariant_metric&oldid=46460
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article