Darboux net invariants

The expressions $h$ and $k$,

$$h=c+ab-\frac{\partial a}{\partial u},\quad k=c+ab-\frac{\partial b}{\partial v}, $$ derived from the coefficients of the Laplace equation (in differential line geometry)

$$\frac{\partial^2\theta}{\partial u\partial v} = a\frac{\partial\theta}{\partial u}+b\frac{\partial\theta}{\partial v}+c\theta. \tag*{(*)}$$ Equation (*) is satisfied by the homogeneous coordinates of a point $x$ describing a conjugate net of lines $u$ and $v$ on a two-dimensional surface in an $n$-dimensional projective space, where $n\geq 3$. It was shown by G. Darboux [1] that the Darboux invariants $h$ and $k$ do not change their value when the normalization of the coordinates of the point $x$ 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 $h$ and $k$ are more commonly referred to as the Darboux invariants of a net.