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Pseudo-totally umbilical lightlike submanifolds

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

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.
Year 2022, Volume: 51 Issue: 5, 1304 - 1320, 01.10.2022
https://doi.org/10.15672/hujms.932487

Abstract

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.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Samuel Ssekajja 0000-0003-4247-1989

Publication Date October 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 5

Cite

APA Ssekajja, S. (2022). Pseudo-totally umbilical lightlike submanifolds. Hacettepe Journal of Mathematics and Statistics, 51(5), 1304-1320. https://doi.org/10.15672/hujms.932487
AMA Ssekajja S. Pseudo-totally umbilical lightlike submanifolds. Hacettepe Journal of Mathematics and Statistics. October 2022;51(5):1304-1320. doi:10.15672/hujms.932487
Chicago Ssekajja, Samuel. “Pseudo-Totally Umbilical Lightlike Submanifolds”. Hacettepe Journal of Mathematics and Statistics 51, no. 5 (October 2022): 1304-20. https://doi.org/10.15672/hujms.932487.
EndNote Ssekajja S (October 1, 2022) Pseudo-totally umbilical lightlike submanifolds. Hacettepe Journal of Mathematics and Statistics 51 5 1304–1320.
IEEE S. Ssekajja, “Pseudo-totally umbilical lightlike submanifolds”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, pp. 1304–1320, 2022, doi: 10.15672/hujms.932487.
ISNAD Ssekajja, Samuel. “Pseudo-Totally Umbilical Lightlike Submanifolds”. Hacettepe Journal of Mathematics and Statistics 51/5 (October 2022), 1304-1320. https://doi.org/10.15672/hujms.932487.
JAMA Ssekajja S. Pseudo-totally umbilical lightlike submanifolds. Hacettepe Journal of Mathematics and Statistics. 2022;51:1304–1320.
MLA Ssekajja, Samuel. “Pseudo-Totally Umbilical Lightlike Submanifolds”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 5, 2022, pp. 1304-20, doi:10.15672/hujms.932487.
Vancouver Ssekajja S. Pseudo-totally umbilical lightlike submanifolds. Hacettepe Journal of Mathematics and Statistics. 2022;51(5):1304-20.