Vol. 15, No. 4, 1965

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Isometric immersions of manifolds of nonnegative constant sectional curvature

Edsel Ford Stiel

Vol. 15 (1965), No. 4, 1415–1419

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.

Mathematical Subject Classification
Primary: 53.74
Received: 5 August 1964
Published: 1 December 1965
Edsel Ford Stiel