# 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 [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