Kyungpook Mathematical Journal 2008; 48(2): 183-194
Published online June 23, 2008
Copyright © Kyungpook Mathematical Journal.
Conformally Flat Totally Umbilical Submanifolds in Some Semi-Riemannian Manifolds
Stanis law Ewert-Krzemieniewski
Institute of Mathematics, Szczecin University of Technology, Al. Piast´ow 17, 70310 Szczecin, Poland
We prove that totally umbilical submanifold $M$ of an extended quasi-recurren manifold is also extended quasi-recurrent. If, moreover, $M$ is conformally flat then, locally, $M$ is isometric to the manifold with known metric. Some curvature properties of such submanifold are investigated. Making use of these results we shall prove the existence of totally umbilical submanifold being pseudosymmetric in the sense of Ryszard Deszcz and satisfying some other curvature conditions.
Keywords: submanifold, recurrent, quasi-recurrent, extended quasirecurrent, conformally flat, totally umbilical, umbilic, Ricci symmetric, Riccipseudosymmetric, pseudosymmetric, pseudosymmetry