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 it is defined as follows.

Suppose one is given a vector field along a curve that is asymptotic (or at least has tangency of the second order with an asymptotic curve at ). The quadric containing three infinitely-close straight lines passing through three points of the curve in the direction of the vectors , where is a frame in and is the affine normal, is called the Lie quadric. Its equation has the form

where together with are the homogeneous coordinates of the curves, is the asymptotic tensor and 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

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

and has contact of the third order with it at ; here is the Gaussian curvature of the tensor .

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)