Namespaces
Variants
Actions

Difference between revisions of "Darboux quadric"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
A second-order surface with second-order contact with a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301501.png" /> in three-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301502.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301503.png" />, in which the line of intersection with the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301504.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301505.png" /> has a special type of singularity. Out of the set of quadrics with second-order contact with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301506.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301507.png" /> one can select the quadrics in which the line of intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301508.png" /> has a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d0301509.png" /> with three coincident tangents. On the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015010.png" /> there are three directions (Darboux directions) for these three coincident tangents. At <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015011.png" /> there exists a one-parameter family of Darboux quadrics — the Darboux pencil. A pencil of hyper-quadrics, which are in contact at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015012.png" /> with a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015013.png" /> in projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015014.png" />, is an extension of the Darboux pencil. A (non-developable) hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030150/d03015015.png" /> degenerates into a hyper-quadric if and only if its generalized [[Darboux tensor|Darboux tensor]] vanishes [[#References|[2]]].
+
{{TEX|done}}
 +
A second-order surface with second-order contact with a surface $S$ in three-dimensional projective space $P_3$ at a point $x$, in which the line of intersection with the surface $S$ at the point $x$ has a special type of singularity. Out of the set of quadrics with second-order contact with $S$ at $x$ one can select the quadrics in which the line of intersection with $S$ has a singular point $x$ with three coincident tangents. On the surface $S$ there are three directions (Darboux directions) for these three coincident tangents. At $x\in S$ there exists a one-parameter family of Darboux quadrics — the Darboux pencil. A pencil of hyper-quadrics, which are in contact at a point $x$ with a hypersurface $S$ in projective space $P_{n+1}$, is an extension of the Darboux pencil. A (non-developable) hypersurface $S$ degenerates into a hyper-quadric if and only if its generalized [[Darboux tensor|Darboux tensor]] vanishes [[#References|[2]]].
  
 
====References====
 
====References====

Revision as of 13:22, 29 April 2014

A second-order surface with second-order contact with a surface $S$ in three-dimensional projective space $P_3$ at a point $x$, in which the line of intersection with the surface $S$ at the point $x$ has a special type of singularity. Out of the set of quadrics with second-order contact with $S$ at $x$ one can select the quadrics in which the line of intersection with $S$ has a singular point $x$ with three coincident tangents. On the surface $S$ there are three directions (Darboux directions) for these three coincident tangents. At $x\in S$ there exists a one-parameter family of Darboux quadrics — the Darboux pencil. A pencil of hyper-quadrics, which are in contact at a point $x$ with a hypersurface $S$ in projective space $P_{n+1}$, is an extension of the Darboux pencil. A (non-developable) hypersurface $S$ degenerates into a hyper-quadric if and only if its generalized Darboux tensor vanishes [2].

References

[1] S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[2] G.F. Laptev, "Differential geometry of imbedded manifolds. Group theoretical methods of differential geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)


Comments

References

[a1] E. Cartan, "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars (1937) pp. Part II, Chapt. VI §II
How to Cite This Entry:
Darboux quadric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_quadric&oldid=14619
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article