Conformally-invariant metric

on a Riemann surface

A rule that associates with each local parameter , mapping a parameter neighbourhood into the closed complex plane (), a real-valued function

such that for all local parameters and for which the intersection is not empty, the following relation holds:

where is the image of in under . A conformally-invariant metric is often denoted by the symbol , to which the indicated invariance with respect to the choice of the local parameter is attributed.

Every linear differential (or quadratic differential ) induces a conformally-invariant metric, (or ). 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 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

References

[a1]	L.V. Ahlfors, "Lectures on quasiconformal mappings" , v. Nostrand (1966)