Math. Z. 173, 111-117 (1980) Mathematische Zeitschrift 9 by Springer-Verlag 1980 Isometric Immersions in Codimension Two with Non Negative Curvature Yuriko Y. Baldin 1 and Francesco Mercuri 2 1 Departamento de Matemfitica Universidade Federal de SgtoCarlos, 13560 S~o Carlos, SP, Brasil 2 Departamento de Matemfitica, Universidade Estadual de Campinas, 13100 Campinas, SP, Brasil Introduction Let f: M"---,1R "+p be an isometric immersion of a compact connected, oriented riemannian manifold of dimension n in an euclidean space of dimension n +p, with non negative sectional curvatures. It is well known that if p= 1, M" is diffeomorphic to a sphere, and in fact, f is an embedding and f(M") is the boundary of a convex body. In [15] Moore proves that if p = 2 and all the sectional curvatures are strictly positive then M" is a homotopy sphere. If p=2 and the sectional curvatures are just non negative, other examples appear besides spheres, namely the product of two spheres. The aim of this paper is to prove that those are essentially the only possibilities. More precisely, we wilt prove that/f the sectional curvatures at some point of M R are all strictly positive then M ~ is homeomorphic to a sphere and, in the general case of non negative curvature, M ~ is either a homotopy sphere or diffeomorphic to a product of two spheres, except for a case in dimension 3 that we were not able to exclude. Moreover, if M" is simply connected and is not homeomorphic to a sphere, M" splits isometrically as product of two manifolds each one isometrically embedded in codimension one and f is the product immersion (and therefore is an embedding). The ideas and techniques used here are largely suggested by Moore's work. 1. Notations and Known Facts For the rest of this paper M will always be an n-dimensional compact, connected, oriented riemannian manifold with non negative sectional curvatures and f: M"---,1R"+2 an isometric immersion. If n = 2 then, by the Gauss Bonnet Theorem, M is either diffeomorphic to a sphere or a flat torus. So we will suppose n__> 3. 0025-5874/80/0173/0111/$01.40