View source for Liouville net

{{TEX|done}}{{MSC|53A05}}

A net of parametrized curves on a surface such that the line element of the surface has the form
$$
ds^2 = (U+V)(du^2 + dv^2)
$$
where $U = U(u)$, $V = V(v)$. In every rectangle formed by two pairs of curves of the different families, the two geodesic diagonals have the same length. Surfaces that carry a Liouville net are [[Liouville surface]]s. For example, central surfaces of the second order are Liouville surfaces. The Liouville net was introduced by [[Joseph Liouville|J. Liouville]] in 1846 (see [[#References|[1]]], Prop. 3).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Monge,   "Application de l'analyse à la géométrie" , Bachelier  (1850)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,   "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French) {{ZBL|0629.53001}}</TD></TR>
</table>