# Darboux surfaces

From Encyclopedia of Mathematics

*wreath of*

Surfaces associated with an infinitesimal deformation of one of them; discovered by G. Darboux [1]. Darboux surfaces form a "wreath" of 12 surfaces, with radius vectors satisfying the equations

where and are in Peterson correspondence, and are in polar correspondence, while and are poles of a -congruence. A similar "wreath" is formed by pairs of isometric surfaces of an elliptic space.

#### References

[1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 4 , Gauthier-Villars (1896) |

[2] | V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian) |

#### Comments

For the notion of a -congruence cf. Congruence of lines.

#### References

[a1] | G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931) |

[a2] | G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954) |

**How to Cite This Entry:**

Darboux surfaces.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=18810

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article