Lie quadric

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)

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)