Pseudo-totally umbilical lightlike submanifolds
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.
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- [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.