Abstract
We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant \({\overline{\phi}}\)-holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.
Similar content being viewed by others
References
Asperti A.C., Mercuri F.: Semi-parallel immersions in space forms. Boll. Un. Mat. Ital. B (7) 8, 883–895 (1994)
Bejancu A.: Umbilical semi-invariant submanifolds of a Sasakian manifold. Tensor N. S. 37, 203–213 (1982)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203, Birkhauser Boston, Inc., Boston (2002)
Bonome A., Castro R., García-Rio E., Hervella L.: Curvature of indefinite almost contact manifolds. J. Geom. 58, 66–86 (1997)
Deprez J.: Semi-parallel surfaces in Euclidean space. J. Geom. 25, 192–200 (1985)
Deprez J.: Semi-parallel hypersurfaces. Rend. Sem. Mat. Univ. Politec. Torino 44, 303–316 (1986)
Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Its Applications. Mathematics and Applications, vol. 364. Kluwer, Dordrecht (1996)
Ferus D.: Immersions with parallel second fundamental form. Math. Z. 140, 87–93 (1974)
Janssens D., Vanhecke L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)
Kenmotsu K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)
Kon M.: Remarks on anti-invariant submanifold of a Sasakian manifold. Tensor, N. S. 30, 239–246 (1976)
Kupeli, D.N.: Singular semi-invariant Geometry. Mathematics and Applications, vol. 366. Kluwer, Dordrecht (1996)
Massamba F.: Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds. Kodai Math. J. 31, 338–358 (2008)
Szabo Z.I.: Structure theorem on Riemannian spaces satisfying R(X, Y).R = 0, I: the local version. J. Differ. Geom. 17, 531–582 (1982)
Szabo Z.I.: Structure theorem on Riemannian spaces satisfying R(X, Y).R = 0, II: the global version. Geom. Dedicata 19, 65–108 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Massamba, F. On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds. J. Geom. 95, 73–89 (2009). https://doi.org/10.1007/s00022-010-0021-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-010-0021-7