Namespaces
Variants
Actions

Darboux tensor

From Encyclopedia of Mathematics
Revision as of 17:24, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A symmetric tensor of valency three,

where are the coefficients of the second fundamental form of the surface, is the Gaussian curvature, and and are their covariant derivatives. G. Darboux [1] was the first to investigate this tensor in special coordinates.

The cubic differential form

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)


Comments

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)
How to Cite This Entry:
Darboux tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_tensor&oldid=18025
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article