Difference between revisions of "Conjugate net"
From Encyclopedia of Mathematics
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| − | 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 [[ | + | 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 $M$ 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)</TD></TR> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4</TD></TR> | |
| − | + | </table> | |
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[[Category:Geometry]] | [[Category:Geometry]] | ||
Latest revision as of 07:52, 17 July 2025
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 $M$ 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) |
| [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=56022
Conjugate net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_net&oldid=56022
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article