# Liouville net

From Encyclopedia of Mathematics

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) Zbl 0629.53001 |

**How to Cite This Entry:**

Liouville net.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Liouville_net&oldid=53155

This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article