I . Introduction
'l'he differential geonirtry of the MOBIUS space has seldom been treated in contemporary pcoinetric literature. Here we only nientiori the papers of 0. KOWALSKI [ell and M. A. AKMBHC [2], [3] whcre l~ibliographics of earlier works cart be found. Both authors provrd general and complicated fundamental theorems, but, a s far as we know, the details of differential geonietry of the n-dimensional MOBIVS space retmirted an almost rindeveloped field. Very detailed investigations only exist for 12 5 3, see W. BLASCIIKE and G . THOMSEN [4], T. TAKASIT [13], hrit fronL the modern conceptrial point of view they are written in a manner partly unsatipfactory.
I n this paper we will investigate imniersions f : Ym --f Sn into the MOBIUS space with the help of E. CARTAN'S method of moving frames. A geometric interpretation and :in cxact version of this method was presented in R. SULARKE and A. SVEC [ll]. the terminology arid results of this paper will be used in the following. The necessary preliminaries aboitt the MOBIUS space are assembled in section 2. Section 3 contains some reniarks on the theory of curves in the MOBIUS space 8". Section 4 is devoted to the genwal theory of iintiiersions f : I'm +Sn; we give a characterization of the spheres