# Conformally-invariant metric

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 ). The definition of a conformally-invariant metric can be carried over to Riemann varieties of arbitrary dimension.

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