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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 [[#References|[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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   "Etude géométrique sur les percussions et le choc des corps"  ''Bull. Sci. Math. Ser. 2'' , '''4'''  (1880)  pp. 126–160</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  pp. 210–233  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,   E. Čech,   "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,   "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.P. Lane,   "A treatise on projective differential geometry" , Univ. Chicago Press  (1942)</TD></TR></table>