JOURNAL BROWSE
Search
Advanced SearchSearch Tips
Conformally Flat Totally Umbilical Submanifolds in Some Semi-Riemannian Manifolds
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 48, Issue 2,  2008, pp.183-194
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2008.48.2.183
 Title & Authors
Conformally Flat Totally Umbilical Submanifolds in Some Semi-Riemannian Manifolds
Ewert-Krzemieniewski, Stanislaw;
  PDF(new window)
 Abstract
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;
 Language
English
 Cited by
 References
1.
M. Belkhelfa, R. Deszcz, M. G logowska, M. Hotlo´s, D. Kowalczyk and L. Verstraelen, On some type of curvature conditions, in: PDEs, Submanifolds and Affine Differential Geometry, Banach Center Publication, 57(2002), 179-194.

2.
M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulg. J. Phys., 15(6)(1988), 526-531.

3.
R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg. Ser. A, 44(1992), 1-34.

4.
U. C. De and S. K. Ghosh, On weakly Ricci symmetric spaces, Publ. Math. Debrecen., 60(1-2)(2002), 201-208.

5.
U. C. De, On generalized Ricci-recurrent manifolds, Tensor ( N.S.), 56(3)(1995), 312-317.

6.
S. Ewert-Krzemieniewski, On some generalisation of recurrent manifolds, Math. Pannonica, 4(2)(1993), 191-203.

7.
S. Ewert-Krzemieniewski, On totally umbilical submanifolds of manifolds with certain recurrent condition imposed on the curvature tensor, Math. Pannonica, 5(1)(1994), 119-130.

8.
S. Ewert-Krzemieniewski, On Totally Umbilical Submanifolds of Conformally Quasi-Recurrent Manifolds, Kyungpook Math. J., 41(2)(2001), 371-380.

9.
S. Ewert-Krzemieniewski, Notes on extended recurrent and extended quasi-recurrent manifolds, Universitatis Iagellonicae Acta Mathematica, 41(2003).

10.
S. Ewert-Krzemieniewski, Totally umbilical submanifolds in some semi-Riemannian manifolds, to appear.

11.
W. Grycak, On generalized curvature tensors and symmetric (0, 2) - tensor with a symmetry condition imposed on the second derivative, Tensor, N. S., 33(1979), 150-152.

12.
Z. Olszak, On totally umbilical submanifolds of Cartan-symmetric manifolds, Demonstratio Math. 11(1)(1976), 95-103.

13.
Z. Olszak, On totally umbilical surfaces immersed in Riemannian conformally recurrent and conformally symmetric spaces, Demonstratio Math., 8(3)(1975), 303-311.

14.
Z. Olszak, On totally umbilical surfaces in some Riemannian manifolds, Colloq. Math., 37(1977), 105-111.

15.
Z. Olszak, On Ricci-recurrent manifolds, Collq. Math., 52(1987), 205-211.

16.
M. Prvanovc, Extended recurrent manifolds, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. (1), 1999, 41-50

17.
M. Prvanovc, Extended recurrent manifolds, translation in Russian Math. (Iz. VUZ) 43(1)(1999), 38-47.