The metrical aspect of the line-sphere transformation

Abstract

The line-sphere transformation of Sophus Lie, first described ill llis relnarkable article, ;sUeber Cortlplexe, insbesolldele Linien und Kugelcomplexe, rnit Anwendung auf die Theorie partieller l)ifferentialgleichungen t dezives its interest frozn two causes. Fil stly, lines and spheres are the simplest follrdimensional geometrical mallifolds with which we have to deal; it is illteresting to establish connections between them. SecoIldly, intersecting lines correspond to tallgent spheles. The consequence usually dlawn from this is that the linesphere transformation falls under the genel al type of coiltact transf{rlllation, and has the beautiful property of carrying asymptotic lines over into lines of ourvature. These facts are of capital importallce when the subject is apploached from the point of view of Pfaff equations. But we may adopt a different pOillt of view, alld then the question naturally arises: Can not the correspolldence of intersecting lines with tangent spheres be made to appear as a special case of sotne more general relation conllecting the distallcess of lines with the allgles of intersection of spheres ? It is the object of the present paper to answer this question affirmative]y, to develop as far as pQssible the relations between the metrical properties of line and sphere systx3ms, alld to bring the two into the closest possible coslnection. :0: Before proceedillg to our task, two words of caution are necessary. Filst, as the angles between spheres appear most naturally through their trigonometric fuIlctions, the corresponding line-functions are found ulost elegantly in non-euclidean space, where distances usually appear in trigonometric form. Secondly, the group of contact transformations that carries lines into lilles and keeps distances invariant depends upon six pararneters, the group that leaves the angles of spheres intact is a ten parameter group, isomorpllic with the quinary orthogonal oIle. We must, therefore, introduce some seemingly arbitrary restrictions, i. e.,