Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

Abstract

A Riemannian manifold is said to be semisymmetric if R(X, Y) · R = 0. A submanifold of Euclidean space which satisfies $$\bar R\left( {X,Y} \right)$$ 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.