Arch. Math., Vol. 60, 296-299 (1993) 0003-889X/93/6003-0296 $ 2.30/'0 9 1993 Birkh/iuser Verlag, Basel Hypersurfaces with a constant support function in spaces of constant sectional curvature By MARCOS DAJCZERand RuY TOJEIRO Introduction. Let J':M n -, Q~ +1 be an oriented connected hypersurface where Qi~+l denotes a complete and simply connected Riemannian manifold of constant sectional curvature c. On the sequel, we assume for simplicity c = 0, i or - 1. The support function with respect to an origin Po ~ Qff + ~ is defined as ?, (x) = (x(x), X(x)5 where N (x) is the given orientation and X (x) = qo(r) grad r is the position vector. Here, r (x) = d (x; Po) denotes the distance function to Po in Q~ + 1 and q) (r) equals r, sin r or sinh r if c = 0, 1 or - 1, respectively. When c ~ 0, consider f composed with the standard umbilical inclusion of either S~ +1 ~ N. "+2 or lI-I"_+l 1 c ~..fl+2, where IL "+2 denotes the standard fiat Lorentzian space. It is easy to see (cf. [1]) that ((f(x)-po, N(x)), if c=0, (1) 7(x) = l(-Po, N(x)), if c 4 = 0, where the inner product is taken in the fiat ambient space. Recently, Hasanis and Koutroufiotis ([4]) classified Euclidean hypersurfaces with a constant support function provided that they are complete. The spherical case was considered in [5]. Inspired on their results we prove the following extension. Theorem 1. Let f: M" -~ Q2+ I be an oriented complete hypersurface with constant support function with respect to Po ~ Q2+ 1. Then, either f (M") = S~' x IR"-" c ~ "+ 1 0 < m < n, with the spherical factor centered at Po or c + 0 and f(M") is umbilical. The method in [4] is global in nature and relies on classical results due to Hartman- Nirenberg and Sacksteder. Our approach is more elementary and goes as follows; based on the Gauss parametrization ([3]) we classify locally in Theorem 2 all hypersurfaces in Q~'+~ with a constant support function showing that there exist an abundance of exam- ples. Then, we check which of those are complete. We conclude this note with a result for Euclidean submanifolds of arbitrary codimen- sion.