• Lungiambudila, Oscar (Departement de Mathematiques et Informatique Faculte des Sciences Universite de Kinshasa (UNIKIN)) ;
  • Massamba, Fortune (School of Mathematics Statistics and Computer Science University of KwaZulu-Natal) ;
  • Tossa, Joel (Institut de Mathematiques et de Sciences Physiques Universite Dabomey-Calavi)
  • Received : 2015.08.17
  • Published : 2016.07.31


The paper deals with lightlike hypersurfaces which are locally symmetric, semi-symmetric and Ricci semi-symmetric in indefinite Kenmotsu manifold having constant $\bar{\phi}$-holomorphic sectional curvature c. We obtain that these hypersurfaces are totally goedesic under certain conditions. The non-existence condition of locally symmetric lightlike hyper-surfaces are given. Some Theorems of specific lightlike hypersurfaces are established. We prove, under a certain condition, that in lightlike hyper-surfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the parallel, semi-parallel, local symmetry, semi-symmetry and Ricci semi-symmetry notions are equivalent.


  1. B. E. Abdalla and R. A. Dillen, A Ricci-semi-symmetric hypersurface of Euclidean space which is not semi-symmetric, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1805-1808.
  2. A. Bejancu, Umbilical semi-invariant submanifolds of a Sasakian manifold, Tensor (N. S.) 37 (1982), no. 1, 203-213.
  3. A. Bonome, R. Castro, E. Garcia-Rio, and L. Hervella, Curvature of indefinite almost contact manifolds, J. Geom. 58 (1997), no. 1-2, 66-86.
  4. C. Calin, Contribution to geometry of CR-submanifold, Ph.D. Thesis, University of Iasi, Iasi, Romania, 1998.
  5. F. Defever, Ricci-semisymmetric hypersurfaces, Balkan J. Geom. Appl. 5 (2000), no. 1, 81-91.
  6. F. Defever, R. Descz, D. Z. Senturk, L. Verstraelem, and S. Yaprak, On a problem of P. J. Ryan, Kyungpook Math. J. 37 (1997), no. 2, 371-376.
  7. F. Defever, R. Descz, D. Z. Senturk, L. Verstraelem, and S. Yaprak, J. Ryan's problem in semi-Riemannian space form, Glasg. Math. J. 41 (1999), no. 2, 271-281.
  8. J. Deprez, Semi-parallel surfaces in Euclidean space, J. Geom. 25 (1985), no. 2, 192-200.
  9. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Amsterdam 1996.
  10. R. Gunes, B. Sahin, and E. Kilic, On lightlike hypersurfaces of semi-Riemannian space form, Turk J. Math. 27 (2003), no. 2, 283-297.
  11. D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), no. 1, 1-27.
  12. K. Kenmotsu, A class of almost contact Riemannian manifold, Tohoku Math. J. 24 (1972), 93-103.
  13. M. Kon, Remarks on anti-invariant submanifold of a Sasakian manifold, Tensor (N. S.) 30 (1976), no. 3, 239-246.
  14. O. Lungiambudila, F. Massamba, and J. Tossa, Symmetry properties of lightlike hypersurfaces in indefinite Sasakian manifolds, SUT J. Math. 46 (2010), no. 2, 177-204.
  15. F. Massamba, On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds, J. Geom. 95 (2009), no. 1-2, 73-89.
  16. F. Massamba, Symmetries of null geometry in indefinite Kenmotsu manifolds, Mediterr. J. Math. 10 (2013), no. 2, 1079-1099.
  17. Y. Matsuyama, Complete hypersurfaces with $R\;{\cdot}\;S\;=\;0\;in\;E^{n+1}$. Proc. Amer. Math. Soc. 88 (1983), no. 1, 119-123.
  18. P. J. Ryan, A class of complex hypersurfaces, Colloq. Math. 26 (1972), 175-182.
  19. B. Sahin, Lightlike hypersurfaces of semi-Euclidean spaces satisfying curvature conditions of semisymmetry type, Turkish J. Math. 31 (2007), no. 2, 139-162.
  20. Z. I. Szabo, Structure theorem on Riemannian spaces satisfying R(X, Y)${\cdot}$R = 0, I: The local version, J. Differential Geom. 17 (1982), no. 4, 531-582.
  21. Z. I. Szabo, Structure theorem on Riemannian spaces satisfying R(X, Y)${\cdot}$R = 0, II: The global version, Geom. Dedicata 19 (1985), no. 1, 65-108.