View source for Darboux tensor

A symmetric tensor of valency three,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301802.png" /> are the coefficients of the second fundamental form of the surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301803.png" /> is the Gaussian curvature, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301805.png" /> are their covariant derivatives. G. Darboux [[#References|[1]]] was the first to investigate this tensor in special coordinates.

The cubic differential form

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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>