Totally Umbilical Screen Transversal Lightlike Submanifolds of Semi-Riemannian Product Manifolds

S. M. Khursheed Haider, Advin Maseih, Mamta Thakur

Department of Mathematics, Jamia Millia Islamia, New Delhi, India.

DOI: 10.4236/apm.2012.24037                

Abstract

We study totally umbilical screen transversal lightlike submanifolds immersed in a semi-Riemannian product manifold and obtain necessary and sufficient conditions for induced connection ▽ on a totally umbilical radical screen transversal lightlike submanifold to be metric connection. We prove a theorem which classifies totally umbilical ST-anti-invariant lightlike submanifold immersed in a semi-Riemannian product manifold.

Keywords

Semi-Riemannian Product Manifolds; Lightlike Submanifolds; Totally Umbilical Radical ST-Lightlike Submanifolds; Totally Umbilical ST-Anti-Invariant Lightlike Submanifolds

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	K. L. Duggal and A. Bejancu, “Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications,” KluwerAcademic Publishers, Dordrecht, 1996.
[2]	D. N. Kupeli, “Singular Semi-Riemannian Geometry,” Kluwer Academic Publishers, Dordrecht, 1996.
[3]	A. Bejancu, “Umbilical CR-submanifolds of a Kaehler Manifold,” Rendiconti di Matematica, Vol. 13, 1980, pp. 431-466.
[4]	S. Deshmukh and S.I. Husain, “Totally Umbilical CR-Submanifolds of a Kaehler Manifold,” Kodai Mathematical Journal, Vol. 9, No. 3, 1986, pp. 425-429.
[5]	doi:10.2996/kmj/1138037271 K. L. Duggal and D. H. Jin, “Totally Umbilical Lightlike Submanifolds,” Kodai Mathematical Journal, Vol. 26, No. 1, 2003, pp. 49-68. doi:10.2996/kmj/1050496648
[6]	M. Gogna, R. Kumar and R. K. Nagaich, “Totally Umbilical CR-lightlike Submanifolds of Indefinite Kaehler Manifolds,” Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 4, 2010, pp. 54-61.
[7]	B. Sahin, “Transversal Lightlike Submanifolds of Indefinite Kaehler Manifolds,” Analele Universitaii de Vest, Timisoara Seria Matematica-Informatica, Vol. XLIV, No. 1, 2006, pp. 119-145.
[8]	C. Yildirim and B. Sahin, “Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds,” Turkish Journal of Mathematics, Vol. 33, 2009, pp. 1-23.
[9]	B. Sahin, “Screen Transversal Lightlike Submanifolds of Indefinite Kaehler Manifolds,” Chaos, Solitons and Fractals, Vol. 38, No. 5, 2008, pp. 1439-1448. doi:10.1016/j.chaos.2008.04.008