Union-preserving transformations of space

Abstract

1. General statement. Soph us Lie studied transformations from lineal-elements into lineal-elements, and also transformations from surface-elements into surface-elements of space. The contact group is obtained by requiring all unions to be converted into unions. Lie's fundamental theorems may be stated as follows. All the contact linealelement transformations form the group of extended point transformations. The contact surface-element transformations which are not merely extended point transformations are defined completely by either a single directrix equation, or a pair of directrix equations. In the first case, a point corresponds to a surface; and in the second case, a point corresponds to a curve. We extend the preceding results by studying transformations in space from differential curve-elements of order (x, y, zf y', z', • • • , y(n)f z(n)^f w h e r e n is 2 or more, into lineal-elements (X> F, Z, Y', Z ' ) . An example of such a transformation arises in the problem of finding the locus of the centers of spherical curvature for an arbitrary space curve. This problem leads to a transformation from curve-elements of third order into lineal-elements. We determine the general class of union-preserving transformations by means of a directrix equation. Lie has obtained directrix equations only for contact transformations of surface-elements since there are no contact transformations of lineal-elements besides the extended point transformations. For a point-to-surface transformation, Lie's standard directrix equation is of the form fl(X, F, Z, x, y, z) = 0. For a point-to-curve transformation, there are two standard directrix equations of the forms Oi(X, F, Z, x, y, s) = 0 , Q2(-X, F, Z, x} y, z) = 0. We find that any general union-preserving transformation from curveelements of order n into lineal-elements is completely determined by our new directrix equation, involving derivatives, 0(X, F, Z, x, y, z, y't* • • • ,y ( n 2 ) , 2 ( )=o. In the final part of our paper, we shall prove that the only available union-preserving transformations (in the whole domain of curvePresented to the Society, November 27, 1943; received by the editors August 17, 1943. We have studied the two-dimensional aspects of our new theory in Proc. Nat. Acad. Sci. (1943) and Revista des Matimaticas (1943). This leads to a large extension of the Huygens theory of evolutes and involutes and Lie theory. x Lie-Scheffers, Beruhrungstransformationen. 2 See Bull. Amer. Math. Soc. abstract 49-9-235 by Kasner and DeCicco, A generalized theory of contact transformations.