Liouville net

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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).


[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) Zbl 0629.53001
How to Cite This Entry:
Liouville net. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article