Pseudo-totally umbilical lightlike submanifolds

Year 2022, Volume: 51 Issue: 5, 1304 - 1320, 01.10.2022

Samuel Ssekajja

https://doi.org/10.15672/hujms.932487

Abstract

We introduce the geometry of pseudo-totally umbilical lightlike submanifold $M$ of a semi-Riemannian manifold $\bar{M}$. In line with the above, we give a complete classification of pseudo-totally umbilical $1$-lightlike submanifolds, such as the lightlike hypersurfaces and half-lightlike submanifolds. Furthermore, pseudo-totally umbilical screen distributions are also investigated, with a complete classification for any lightlike hypersurfaces whose screen distributions are pseudo-totally umbilical. Closely linked to the above we also show, under some geometric conditions, that some pseudo-totally umbilical leaves $M^{*}$, of the screen distribution over $M$, as non-degenerate submanifolds of $\bar{M}$, are either contained in semi-Euclidean spheres or the hyperbolic spaces. Moreover, tangible examples are constructed in this case. Finally, we introduce the notion of mean lightlike sectional curvatures and relate them to the well-known tensors used in the characterisation of lightlike hypersurfaces in space-times.

Keywords

lightlike submanifolds, pseudo-totally umbilical submanifolds, mean lightlike sectional curvatures

Supporting Institution

University of the Witwatersrand

References

• [1] C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Balkan Society of Geometers, Applied Sciences, 11, 9–18, 2009.
• [2] C. Atindogbe and K.L. Duggal, Conformal screen on lightlike hypersurfaces. Int. J. Pure Appl. Math. 11(4), 421–442, 2004.
• [3] C. Calin, Contributions to geometry of CR-submanifold, Thesis, University of Iasi- Romania, 1998.
• [4] B.Y. Chen, Finite type submanifolds in pseudo-euclidean spaces and applications, Kodai Math. J. 8, 358–374, 1985.
• [5] B.Y. Chen, Some results of Chern-do carmo-Kobayashi type and the length of second fundamental form, Indiana Univ. Math. J. 20, 1175–1185, 1971.
• [6] B.Y. Chen, Pseudo-umbilical submanifolds of a Riemannian manifold of constant curvature, II, J. Math. Soc. Japan, 25(1), 105–114, 1973.
• [7] B.Y. Chen and K. Yano, Pseudo-umbilical submanifolds of a Riemannian manifold of constant curvature, Differential Geometry, in honour of K. Yano, Tokyo, 61–71, 1972.
• [8] K.L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and Its Applications, vol. 364, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
• [9] K.L. Duggal and D.H. Jin, Totally umbilical lightlike submanifolds. Kodai Math. J. 26, 49–68, 2003.
• [10] K.L. Duggal and D.H. Jin, Lightlike curves and hypersurfaces of semi-Riemannian manifolds, World Scientific Publishing, Hackensack, NJ, USA, 2007.
• [11] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhauser, Basel, Switzerland, 2010. .
• [12] K.L. Duggal and B. Sahin, Lightlike submanifolds of indefinite Sasakian manifolds, Int. J. Math. Math. Sci. 2007, Article ID 57585, 21 pages, 2007.
• [13] R.S. Gupta and A. Sharfuddin, Generalised Cauchy-Riemann lightlike submanifolds of indefinite Kenmotsu manifolds, Note Mat. 30(2), 49–59, 2010.
• [14] S. Huafei, Pseudo-umbilical submanifolds of a space form $N^{n+p}(C)$, Tsukuba J. Math. 20(1), 45–50, 1996.
• [15] D.N. Kupeli, Singular semi-Riemannian geometry, Mathematics and Its Applications, Vol. 366, Kluwer Academic Publishers, Dordrecht, 1996.
• [16] B. Sahin and C. Yildirim, Slant lightlike submanifolds of indefinite Sasakian manifolds, Filomat, 26(2), 277–287, 2012.
• [17] B. Sahin, Screen transversal lightlike submanifolds of Indefinite Kaehler Manifolds, Chaos Solitons Fractals, 38, 1439–1448, 2008.
• [18] S. Ssekajja, Some results on null hypersurfaces in $(LCS)$-manifolds, Kyungpook Math. J. 59, 783–795, 2019.
• [19] S. Ssekajja, A note on null hypersurfaces of indefinite Kaehler space forms, Balkan J. Geom. Appl. 25(2), 94–105, 2020.
• [20] S. Ssekajja, Geometry of isoparametric null hypersurfaces of Lorentzian manifolds, J. Korean Math. Soc. 57(1), 195–213, 2020.