Namespaces
Variants
Actions

Difference between revisions of "Lie quadric"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (gather refs)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
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.
+
{{TEX|done}}
 +
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
+
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>
+
$$g_{ij}\xi^i\xi^j+H\xi\xi-2\xi=0,$$
  
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.
+
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>
+
$$g_{ij}\xi^i\xi^j+\kappa\xi\xi-2\xi=0,$$
  
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
+
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>
+
$$g_{ij}\xi^i\xi^j+\frac23(H+\kappa)\xi\xi-2\xi=0,$$
  
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" />.
+
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]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lie, ''Gesammelte Abhandlungen. Anmerkungen zum 3-ten Band'' , Teubner (1922)  pp. 718</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.A. Shirokov,   A.P. Shirokov,  "Affine differential geometry" , Moscow (1959) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,   "Projective differential geometry" , Moscow-Leningrad  (1937) (In Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lie, ''Gesammelte Abhandlungen. Anmerkungen zum 3-ten Band'', Teubner (1922)  pp. 718</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.A. Shirokov, A.P. Shirokov,  "Affine differential geometry", Moscow (1959) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov, "Projective differential geometry", Moscow-Leningrad  (1937) (In Russian)</TD></TR>
 
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR>
 
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , '''2''' , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , '''2''' , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>
 

Latest revision as of 08:52, 8 April 2023

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)
[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