Sophus Lie, the mathematician
Sophus Lie, the mathematician
Sophus Lie, the mathematician
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line geometry and comes up with <strong>the</strong> number 16 but here he is talkingabout spheres whose coordinates are complex.Since <strong>the</strong> line-sphere transformation necessarily passes through <strong>the</strong>complex numbers it has not really found its rightful place in moderndifferential geometry. Of <strong>the</strong> major classic works on <strong>the</strong> subject it seemsthat only <strong>the</strong> books by Darboux and Blaschke treat it, in fact muchof Blaschke's third volume from 1929 is devoted to it. For a moderntreatment see Cecil's book [10].<strong>Lie</strong> in his <strong>the</strong>sis, and in <strong>the</strong> above-mentioned expansion <strong>the</strong>reof, embarksalso on a general <strong>the</strong>ory of contact-transformations, transformationsof <strong>the</strong> cotangent bundle T*(C n ) into itself preserving <strong>the</strong> canonical1-form (<strong>the</strong> line-sphere transformation being an example) In <strong>the</strong> usualcoordinates (l,... ,zpnl,..., pn) on T*(Cn), a contact transformationis an isomorphism (i,pi) --* (Xi, Pi) such that Epii = E PiXi.Such transformations appear in <strong>the</strong> integration of first order partialdifferential equationsF axl'"" xl,...,z,~, ax, = o (1)In this context consider <strong>the</strong> Poisson bracket of functionsand vector fieldsand <strong>the</strong> <strong>Lie</strong> bracket(f g)= (aiapi a ap (2)X= aj Y= E bk (3)jkbk .oak a[X Y = (aj bj aXx a (4)jk - a k -X<strong>Lie</strong> views <strong>the</strong> Poisson bracket as <strong>the</strong> effect on f of a vector field Xgassociated with g and interprets <strong>the</strong> Jacobi identity for (f, g) as <strong>the</strong> factthat <strong>the</strong> bracket of <strong>the</strong> operators corresponding to g and h is associatedto (g, h):[X9, Xh] = X(g,h)Thus <strong>the</strong> first order differential equation for g