Sophus Lie, the mathematician

IIt should however be remembered that the mapping is based on complexifyingsome of the coordinates. In fact a surface may not have anyasymptotic lines (they are in other words imaginary). Thus the surfacetheorem above does not figure prominently in modern differential geometry.In fact, Blaschke' s work [2], Vol. 3, is the last one that I haveseen that treats it.However, the transformation has a certain "real form" (see e.g. Cecil[10]): Consider the quadricQ 4 : _ 2 + 2 + 2 = 0in real projective space P 5 . This quadric contains projective lines but nolinear subspaces of higher dimension. The mapping of Lie is a bijectionbetween the set of oriented spheres, oriented planes and point spheresin R 3 U 0o onto the set of points on Q4 . Let denote the dot producton R 3 .R; 3 U Coo} Sr(a): Sphere, center a, radius rQ 4 P(a,r): Point (1 + aa-r 2 ,1-a +ra , 2 2al,2a2,2a3,r).I, The mapping Sr (a) - P(a, r) is a combination of a stereographic projectionand an imbedding of R 4 into P 4 and of R 5 into P 5 . Under thismap a family of spheres with the same tangent plane at a point getsmapped to a line on Q 4 . Lie' also proved that any line-preserving diffeomorphismof Q 4 is the restriction to Q 4 of a projective transformation ofP 5 , i.e. a member of PO(4, 2). Since the signature of the quadric P = 0in line geometry was (3,3) and since the groups PO(3, 3) and PO(4, 2)are not locally isomorphic, real line geometry and real sphere geometryare different. However the groups have the same complexification.As mentioned, Lie made this discovery in early July 1870. Klein,who lived in an adjoining room in the hotel recalls (cf. [19], Vol. 1,p. 97):"... one morning I got up early and wanted to go out rightaway when Lie, who still lay in bed called me into his room.He explained tome the relationship he had found during thenight between the .asymptotic curves of one surface and the"lines of curvature of another, but in such a way that I couldnot understand a word. In any case, he assured me thatS