Namespaces
Variants
Actions

Conjugate net

From Encyclopedia of Mathematics
Revision as of 17:15, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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