On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures

THEOREM 1.1. Assume that (i) Md is a d-dimensional (d_ 2), complete Riemann manifold of class Ck, k> 2, with nonnegative sectional curvatures; (ii)f: Md -> Ed +e is a Ck isometric immersion of Md into a Euclidean space Ed + e, e >0; and (iii) the relative nullity v(p) of the immersion f at the point p C Md satisfies v(p) > Ofor all p. Let m=min v(p) for p C Md. Then f(Md) is Ckl nm-cylindrical; that is, there exists a complete Riemann manifold Md-m of class Ck-l such that Md, Ed+e, f have factorizations Md=Emx Md-m Ed+ e=EmxEd+ em, f=ixg, where i: Em ->Em is the identity map and g: Md-m Ed+ ermis a Ck-1 isometric immersion. Also, f(Md) is not (m + 1)-cylindrical.