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Conjugate directions

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A pair of directions emanating from a point $ P $ on a surface $ S $ such that the straight lines containing them are conjugate diameters of the Dupin indicatrix of $ S $ at $ P $. In order that the directions $ ( du : dv) $, $ ( \delta u : \delta v) $ at a point $ P $ on $ S $ be conjugate, it is necessary and sufficient that the following condition holds

$$ L du \delta u + M ( du \delta v + dv \delta u) + N dv \delta v = 0, $$

where $ L $, $ M $ and $ N $ are the coefficients of the second fundamental form of $ S $ evaluated at $ P $. Example: a principal direction.

References

[1] A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[a1] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , 1 , Springer (1973)
[a2] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Conjugate directions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_directions&oldid=53806
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article