Darboux net invariants

The expressions and ,

derived from the coefficients of the Laplace equation (in differential line geometry)

(*)

Equation (*) is satisfied by the homogeneous coordinates of a point describing a conjugate net of lines and on a two-dimensional surface in an -dimensional projective space, where . It was shown by G. Darboux [1] that the Darboux invariants and do not change their value when the normalization of the coordinates of the point is changed. Special forms of conjugate nets are obtained by imposing some condition on the Darboux invariants.

References

[1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 2 , Gauthier-Villars (1889)
[2]	G. Tzitzeica, "Géométrie différentielle projective des réseaux" , Gauthier-Villars & Acad. Roumaine (1924)
[3]	S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)

Comments

The expressions and are more commonly referred to as the Darboux invariants of a net.