Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

ISSN： 0011-4642

Source： Czechoslovak Mathematical Journal, Vol.53, Iss.3, 2003-09, pp. : 707-734

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Abstract

A Riemannian manifold is said to be semisymmetric if R(X, Y) · R = 0. A submanifold of Euclidean space which satisfies $\bar r(x,y)\cdot h= 0$ is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played by the foliated ones, which in the dimension 3 are divided by Kowalski into four classes: elliptic, hyperbolic, parabolic and planar. It is shown now that only the planar ones can be immersed isometrically into Euclidean spaces as 3-dimensional semiparallel submanifolds. This result is obtained by a complete classification of such submanifolds.