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One of the osculating quadrics (cf. [[Osculating quadric|Osculating quadric]]) to a surface in the geometry of an equi-affine or projective group. At a hyperbolic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587301.png" /> it is defined as follows.
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One of the osculating quadrics (cf. [[Osculating quadric|Osculating quadric]]) to a surface in the geometry of an equi-affine or projective group. At a hyperbolic point $M_0$ it is defined as follows.
  
Suppose one is given a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587302.png" /> along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587303.png" /> that is asymptotic (or at least has tangency of the second order with an asymptotic curve at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587304.png" />). The quadric containing three infinitely-close straight lines passing through three points of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587305.png" /> in the direction of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587307.png" /> is a frame in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l0587309.png" /> is the [[Affine normal|affine normal]], is called the Lie quadric. Its equation has the form
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Suppose one is given a vector field $v^i(t)$ along a curve $L=u^i(t)$ that is asymptotic (or at least has tangency of the second order with an asymptotic curve at $M_0$). The quadric containing three infinitely-close straight lines passing through three points of the curve $u^i(t)$ in the direction of the vectors $v^i(t)r_i(t)=c^ir_i+cN$, where $r_1,r_2,N$ is a frame in $M$ and $N$ is the [[Affine normal|affine normal]], is called the Lie quadric. Its equation has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873010.png" /></td> </tr></table>
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$$g_{ij}\xi^i\xi^j+H\xi\xi-2\xi=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873011.png" /> together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873012.png" /> are the homogeneous coordinates of the curves, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873013.png" /> is the asymptotic tensor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873014.png" /> is the affine mean curvature.
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where $(\xi^1:\xi^2:\xi:1)$ together with $(c^1:c^2:c:1)$ are the homogeneous coordinates of the curves, $g_{ij}$ is the asymptotic tensor and $H$ is the affine mean curvature.
  
 
The Lie quadric (together with the Wilczynski and Fubini quadric) belongs to a pencil of Darboux quadrics (cf. [[Darboux quadric|Darboux quadric]]). The first has the equation
 
The Lie quadric (together with the Wilczynski and Fubini quadric) belongs to a pencil of Darboux quadrics (cf. [[Darboux quadric|Darboux quadric]]). The first has the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873015.png" /></td> </tr></table>
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$$g_{ij}\xi^i\xi^j+\kappa\xi\xi-2\xi=0,$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873016.png" /> is a geodesic of the first kind for it, and the second has the equation
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and $L$ is a geodesic of the first kind for it, and the second has the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873017.png" /></td> </tr></table>
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$$g_{ij}\xi^i\xi^j+\frac23(H+\kappa)\xi\xi-2\xi=0,$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873018.png" /> has contact of the third order with it at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873019.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873020.png" /> is the Gaussian curvature of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058730/l05873021.png" />.
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and $L$ has contact of the third order with it at $M_0$; here $\kappa$ is the Gaussian curvature of the tensor $g_{ij}$.
  
 
The idea of the Lie quadric was introduced in a letter from S. Lie to F. Klein on 18 December 1878 (see [[#References|[1]]]).
 
The idea of the Lie quadric was introduced in a letter from S. Lie to F. Klein on 18 December 1878 (see [[#References|[1]]]).

Revision as of 15:08, 27 August 2014

One of the osculating quadrics (cf. Osculating quadric) to a surface in the geometry of an equi-affine or projective group. At a hyperbolic point $M_0$ it is defined as follows.

Suppose one is given a vector field $v^i(t)$ along a curve $L=u^i(t)$ that is asymptotic (or at least has tangency of the second order with an asymptotic curve at $M_0$). The quadric containing three infinitely-close straight lines passing through three points of the curve $u^i(t)$ in the direction of the vectors $v^i(t)r_i(t)=c^ir_i+cN$, where $r_1,r_2,N$ is a frame in $M$ and $N$ is the affine normal, is called the Lie quadric. Its equation has the form

$$g_{ij}\xi^i\xi^j+H\xi\xi-2\xi=0,$$

where $(\xi^1:\xi^2:\xi:1)$ together with $(c^1:c^2:c:1)$ are the homogeneous coordinates of the curves, $g_{ij}$ is the asymptotic tensor and $H$ is the affine mean curvature.

The Lie quadric (together with the Wilczynski and Fubini quadric) belongs to a pencil of Darboux quadrics (cf. Darboux quadric). The first has the equation

$$g_{ij}\xi^i\xi^j+\kappa\xi\xi-2\xi=0,$$

and $L$ is a geodesic of the first kind for it, and the second has the equation

$$g_{ij}\xi^i\xi^j+\frac23(H+\kappa)\xi\xi-2\xi=0,$$

and $L$ has contact of the third order with it at $M_0$; here $\kappa$ is the Gaussian curvature of the tensor $g_{ij}$.

The idea of the Lie quadric was introduced in a letter from S. Lie to F. Klein on 18 December 1878 (see [1]).

References

[1] S. Lie, , Gesammelte Abhandlungen. Anmerkungen zum 3-ten Band , Teubner (1922) pp. 718
[2] P.A. Shirokov, A.P. Shirokov, "Affine differential geometry" , Moscow (1959) (In Russian)
[3] S.P. Finikov, "Projective differential geometry" , Moscow-Leningrad (1937) (In Russian)


Comments

References

[a1] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , 2 , Springer (1923)
[a2] G. Bol, "Projective Differentialgeometrie" , 2 , Vandenhoeck & Ruprecht (1954)
How to Cite This Entry:
Lie quadric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_quadric&oldid=14061
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article