# Conjugate net

From Encyclopedia of Mathematics

A net of lines on a surface consisting of two families of lines such that at every point of the surface the lines from the two families of the net have conjugate directions. If a coordinate net is a conjugate net, then the coefficient of the second fundamental form of the surface is identically equal to zero. In a neighbourhood of every point of the surface which is not a flat point one can introduce a parametrization such that the coordinate lines form a conjugate net. One family can be chosen arbitrarily, even when the lines of this family do not have asymptotic directions. An important example is a net of lines of curvature.

#### References

[1] | A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian) |

#### Comments

#### References

[a1] | C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4 |

**How to Cite This Entry:**

Conjugate net.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Conjugate_net&oldid=16148

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