Let Md denote a C∞
Riemannian manifold which is d-dimensional and complete. Our first result states
that an isometric immersion of a flat Md into (d + k)-dimensional Euclidean space,
k < d, is n-cylindrical if the relative nullity of the immersion has constant value
n. This result was obtained by O’Neill with the additional hypothesis of
vanishing relative curvature. We next consider the case in which Md and
Md+k, k < d, are manifolds of the same constant positive sectional curvature. In
this case we show that an isometric immersion of Md into Md+k is totally
geodesic if the relative curvature of the immersion is zero on a certain subset of
Md.
