Conjugate directions

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A pair of directions emanating from a point on a surface such that the straight lines containing them are conjugate diameters of the Dupin indicatrix of at . In order that the directions , at a point on be conjugate, it is necessary and sufficient that the following condition holds

where , and are the coefficients of the second fundamental form of evaluated at . Example: a principal direction.


[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. E.V. Shikin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098