Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds
Tóm tắt
This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under
$${\overline{\phi}}$$
.
Tài liệu tham khảo
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)
Blair D.E., Oubina J.A.: Conformal and related changes of metric on the product of two almost contact metric manifolds. Publ. Math. 34(1), 199–207 (1990)
Bonome A., Castro R., García-Rio E., Hervella L.: Curvature of indefinite almost contact manifolds. J. Geom. 58, 66–86 (1997)
Brunetti L., Pastore A.M.: Lightlike hypersurfaces in indefinite \({\mathcal{S}}\) -manifolds. Differ. Geom. Dyn. Syst. 12, 18–40 (2010)
Calin C., Mihai I.: On a normal contact metric manifold. Kyungpook Math. J. 45, 55–65 (2005)
Chern S.S.: Pseudo-groupes continus infinis. Colloques Internat. Centre Nat. Rech. Sci. 52, 119–136 (1953)
Cho J.T., Inoguchi J.-I.: On \({\varphi}\) -Einstein contact Riemannian manifolds. Mediterr. J. Math. 7, 143–167 (2010)
De U.C., Tripathi M.M.: Ricci tensor in 3-dimensional trans-Sasakian manifolds. Kyungpook Math. J. 43, 247–255 (2003)
Defever F.: Ricci-semisymmetric hypersurfaces. Balkan J. Geom. Appl. 5(1), 81–91 (2000)
Duggal K.L.: Lorentzian geometry of globally framed manifolds. Acta Appl. Math. 19(2), 131–148 (1990)
Duggal, K.L., Bejancu, A.: Lightlike submanifolds of semi-Riemannian manifolds and applications. In: Mathematics and its Application. Kluwer Publishers, Dordrecht (1996)
Duggal K.L., Jin D.H.: Null Curves and Hypersurfaces of Semi-Riemannian Manifolds. World Scientific Publishing Co. Pvt. Ltd, Singapore (2007)
Duggal, K.L., Sahin, B.: Lightlike submanifolds of indefinite Sasakian manifolds. Int. J. Math. Math. Sci. 2007, Art. ID 57585 (2007)
Gray J.W.: Some global properties of contact structures. Ann. Math. 69(2), 421–450 (1959)
Gray A., Hervella L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Math. Pure Appl. 123(4), 35–58 (1980)
Günes R., Sahin B., Kilic E.: On lightlike hypersurfaces of a semi-Riemannian space form. Turk. J. Math. 27, 283–297 (2003)
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)
Kulkarni R.S.: The values of sectional curvatures in indefinite metrics. Comment. Math. Helv. 54, 173–176 (1979)
Kupeli, D.N.: Singular semi-invariant Geometry. In: Mathematics and Applications, vol. 366. Kluwer Academic Publishers, Dordrecht (1996)
Marrero J.C.: The local structure of trans-Sasakian manifolds. Ann. Math. Pure Appl. 162(4), 77–86 (1992)
Massamba F.: Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilinear forms. Differ. Geom. Dyn. Syst. 10, 226–234 (2008)
Massamba F.: On weakly Ricci symmetric lightlike hypersurfaces of indefinite Sasakian manifolds. SUT J. Math. 44(2), 181–201 (2008)
Massamba F.: Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds. Kodai Math. J. 31(3), 338–358 (2008)
Massamba F.: On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds. J. Geom. 95, 73–89 (2009)
Massamba, F.: On lightlike geometry in indefinite Kenmotsu manifolds. Slovaca Math. (in press, 2011)
Massamba F.: Screen integrable lightlike hypersurfaces of indefinite Sasakian manifolds. Mediterr. J. Math. 6, 27–46 (2009)
Massamba F.: Relative nullity and lightlike hypersurfaces in indefinite Kenmotsu manifolds. Turk. J. Math. 35, 129–149 (2011)
O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Pure Appl Math, vol. 103. Academic Press, New York (1983)
Oubina J.A.: New classes of almost contact metric structures. Publ. Math. Debrecen 32(3–4), 187–193 (1985)
Sasaki S.: On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tôhoku Math. J. (2) 12(3), 459–476 (1960)
Vergara-Diaz E., Wood C.M.: Harmonic almost contact structures. Geom. Dedicata 123, 131–151 (2006)