Geodesic
Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 707-734

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A Riemannian manifold is said to be semisymmetric if $R(X,Y)\cdot R=0$. 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.
Classification : 53B25, 53C25, 53C42
Keywords: semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds 
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     title = {Semiparallel isometric immersions of 3-dimensional semisymmetric {Riemannian} manifolds},
     journal = {Czechoslovak Mathematical Journal},
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Lumiste, Ülo. Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 707-734. http://geodesic.mathdoc.fr/item/CMJ_2003__53_3_a16/