Darboux tensor

A symmetric tensor of valency three,

$$ \theta _ {\alpha \beta \gamma } = b _ {\alpha \beta \gamma } - \frac{b _ {\alpha \beta } K _ \gamma + b _ {\beta \gamma } K _ \alpha + b _ {\gamma \alpha } K _ \beta }{4K} , $$

where $ b _ {\alpha \beta } $ are the coefficients of the second fundamental form of the surface, $ K $ is the Gaussian curvature, and $ b _ {\alpha \beta \gamma } $ and $ K _ \alpha $ are their covariant derivatives. G. Darboux [1] was the first to investigate this tensor in special coordinates.

The cubic differential form

$$ \theta _ {\alpha \beta \gamma } du ^ \alpha du ^ \beta du ^ \gamma = b _ {\alpha \beta \gamma } du ^ \alpha du ^ \beta du ^ \gamma + $$

$$ - \frac{3}{4} \frac{K _ \gamma }{K} b _ {\alpha \beta } du ^ \alpha du ^ \beta du ^ \gamma $$

is connected with the Darboux tensor. This form, evaluated for a curve on a surface, is known as the Darboux invariant. On a surface of constant negative curvature the Darboux invariant coincides with the differential parameter on any one of its curves. A curve at each point of which the Darboux invariant vanishes is known as a Darboux curve. Only one real family of Darboux curves exists on a non-ruled surface of negative curvature. Three real families of Darboux curves exist on a surface of positive curvature. A surface at each point of which the Darboux tensor is defined and vanishes identically is called a Darboux surface. Darboux surfaces are second-order surfaces which are not developable on a plane.

References

[1]	G. Darboux, "Etude géométrique sur les percussions et le choc des corps" Bull. Sci. Math. Ser. 2 , 4 (1880) pp. 126–160
[2]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) pp. 210–233 (In Russian)

References

[a1]	G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931)
[a2]	G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954)
[a3]	E.P. Lane, "A treatise on projective differential geometry" , Univ. Chicago Press (1942)