Liouville net

2020 Mathematics Subject Classification: Primary: 53A05 [MSN][ZBL]

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 surfaces. For example, central surfaces of the second order are Liouville surfaces. The Liouville net was introduced by J. Liouville in 1846 (see [1], Prop. 3).

References

[1]	G. Monge, "Application de l'analyse à la géométrie" , Bachelier (1850)
[2]	V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)