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Difference between revisions of "Conjugate net"

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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 [[Conjugate directions|conjugate directions]]. If a coordinate net is a conjugate net, then the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025060/c0250601.png" /> 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.
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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 [[Conjugate directions|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.
  
 
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<table><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>
 
<table><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]]

Latest revision as of 12:41, 2 November 2014

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)


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