Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds

Ülo Lumiste

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 3, page 707-734
  • ISSN: 0011-4642

Abstract

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A Riemannian manifold is said to be semisymmetric if R ( X , Y ) · R = 0 . A submanifold of Euclidean space which satisfies R ¯ ( X , Y ) · 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.

How to cite

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Lumiste, Ülo. "Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds." Czechoslovak Mathematical Journal 53.3 (2003): 707-734. <http://eudml.org/doc/30810>.

@article{Lumiste2003,
abstract = {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.},
author = {Lumiste, Ülo},
journal = {Czechoslovak Mathematical Journal},
keywords = {semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds; semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds},
language = {eng},
number = {3},
pages = {707-734},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds},
url = {http://eudml.org/doc/30810},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Lumiste, Ülo
TI - Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 3
SP - 707
EP - 734
AB - 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.
LA - eng
KW - semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds; semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds
UR - http://eudml.org/doc/30810
ER -

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