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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)/: Md -> Ed + e is a Ck isometric immersion of Md into a Euclidean space Ed + e, e>0; and (Hi) the relative nullity v(p) of the immersion f at the point p e Md satisfies v(p) > Ofor all p. Let w = min v(p)forp e Md. Then f(Md) is Ck~1 m-cylindrical; that is, there exists a complete Riemann manifold Md~m of class Ck~1 such that Md, Ed + e, f have factorizations Md = EmxMd-m, Ed + e = Emx Ed + e-m, f=ixg, where i: Em-> Em is the identity map and g: Md~m -> Ed + e~m is a Ck~1 isometric immersion. Also, f(Md) is not (m+ l)-cy/indrical.

On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures. II