Comment. Math. Helvetici 63 (1988) 672-674 0010-2571/88/040672-03501.50 + 0.20/0 9 1988 Birkh/iuser Verlag, Basel A non-immersion theorem for spaceforms F. J. PEDIT llllroduction This note gives a uniform treatment of some non-existence results concerning isometric immersions of spaceforms into spaceforms. Let M, A~ be spaceforms of dim M = m, dim A~ = 2m - 1 and curvatures c, 6. Then we prove the following THEOREM. Let M be complete. Then there exists no isometric immersion [ : M-," f4 if (1) c < e, c < 0 and M is fuchsian, (2) 0<c<~. A hyperbolic spaceform is called fuchsian if the limit set of the fundamental group Hi(M) in the sphere at infinity of the universal cover contains more than two (and hence infinitely many) points [2]. Part (1) of our result extends recent work of F. Xavier [6] where the same statement is proven by a different method in case A~ = R ~-' with the standard flat metric. Since every compact hyperbolic spaceform is fuchsian our result also relates to early work of S. S. Chern and N. H. Kuiper [1] where, in order to use compactness of M, A~ has to be diffeomorphic to R ~-1. Behind all of this one clearly has in mind the problem of whether m-dimensional hyperbolic space H m can be isometrically immersed into R z''-l. The case m = 2 was treated by D. Hilbert who showed that there are no isometric immersions of H z into /I~ 3 [4]. Part (2) proves again a result of J. D. Moore [3] and we only add it since it fits canonically into our approach. It is well known [4] that in the case c -> 6 one always has isometric immersions f : M ~/~ for complete M. Our method is based on the observation that an isometric immersion f :M--*At always comes with a flat metric if ~ > c. If M is complete this metric will be complete and hence put restrictions on the fundamental group of M. Basic facts Let 1, ] be the riemannian metrics of constant curvatures c, ~ on M, ,~ and assume x = t~ - c > 0. If f : M ~ A~ is an isometric immersion then f*] = 1 and we 672