On total mean curvatures of foliated half-lightlike submanifolds in semi-Riemannian manifolds

Abstract

We derive total mean curvature integration formulas of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2010].

Keywords

• Half-lightlike submanifold,Newton transformation,foliation and mean curvature

References

1. [1] K. Andrzejewski, W. Kozlowski and K. Niedzialomski, Generalized Newton transformation and its applications to extrinsic geometry, Asian J. Math. 20 (2), 293–322, 2016.
2. [2] K. Andrzejewski and Pawel G. Walczak, The Newton transformation and new integral formulae for foliated manifolds, Ann. Glob. Anal. Geom. 37 (2), 103–111, 2010.
3. [3] C. Calin, Contributions to geometry of CR-submanifold, Ph.D. thesis, University of Iasi (Romania), 1998.
4. [4] K.L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and Its Applications, Kluwer Academic Publishers, 1996.
5. [5] K.L. Duggal and B. Sahin, Screen conformal half-lightlike submanifolds, Int. J. Math. Math. Sci. 2004 (68), 3737–3753, 2004.
6. [6] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2010.
7. [7] D.N. Kupeli, Singuler semi-Riemannian geometry, Mathematics and Its Applications, Vol. 366, Kluwer Academic Publishers, 1996.
8. [8] F. Massamba, Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds, Kodai Math. J. 31, 338–358, 2008.

9. [9] F. Massamba, On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds, J. Geom. 95, 73–89, 2009.
10. [10] F. Massamba and S. Ssekajja, Some remarks on quasi generalized CR-null geometry in indefinite nearly cosymplectic manifolds, Int. J. Math. Math. Sci. Art. ID 9613182, 10 pp, 2016.
11. [11] E. Yasar, A.C. Coken, and A. Yucesan, Lightlike hypersurfaces in semi-Riemannian manifold with semi-symmetric non-metric connection, Math. Scand. 102 (2), 253– 264, 2008.