Revista de la
Unión Matemática Argentina
Warped product lightlike submanifolds with a slant factor
Megha Pruthi and Sangeet Kumar

Volume 68, no. 1 (2025), pp. 1–21    

Published online (final version): December 18, 2024

https://doi.org/10.33044/revuma.3335

Download PDF

Abstract

In the present study, we investigate a new type of warped products on manifolds with indefinite metrics, namely, warped product lightlike submanifolds of indefinite Kaehler manifolds with a slant factor. First, we show that indefinite Kaehler manifolds do not admit any proper warped product semi-slant lightlike submanifolds of the type $N_{T}\times_{\lambda}N_{\theta}$, $N_{\theta}\times_{\lambda}N_{T}$, $N_{\perp}\times_{\lambda}N_{\theta}$ and $N_{\theta}\times_{\lambda}N_{\perp}$, where $N_{T}$ is a holomorphic submanifold, $N_{\perp}$ is a totally real submanifold and $N_{\theta}$ is a proper slant submanifold. Then, we study warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$, where $B = N_{T}\times N_{\perp}$, of an indefinite Kaehler manifold. Following this, we give one non-trivial example for this kind of warped products of indefinite Kaehler manifolds. Then, we establish a geometric estimate for the squared norm of the second fundamental form involving the Hessian of warping function $\lambda$ for this class of warped products. Finally, we present a sharp geometric inequality for the squared norm of second fundamental form of warped product semi-slant lightlike submanifolds of the type $B\times_{\lambda}N_{\theta}$.

References

  1. M. Barros and A. Romero, Indefinite Kähler manifolds, Math. Ann. 261 no. 1 (1982), 55–62.  DOI  MR  Zbl
  2. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49.  DOI  MR  Zbl
  3. A. Carriazo, Bi-slant immersions, in Proc. ICRAMS (Kharagpur, 2000), Narosa, New Delhi, 2001, pp. 88–97.
  4. B.-Y. Chen, Geometry of slant submanifolds, Katholieke Universiteit Leuven, Louvain, 1990. Available as arXiv:1307.1512v2 [math.DG]  MR  Zbl.
  5. B.-Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 no. 1 (1990), 135–147.  DOI  MR  Zbl
  6. B.-Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math. 133 no. 3 (2001), 177–195.  DOI  MR  Zbl
  7. B.-Y. Chen, S. Uddin, and F. R. Al-Solamy, Geometry of pointwise CR-slant warped products in Kaehler manifolds, Rev. Un. Mat. Argentina 61 no. 2 (2020), 353–365.  DOI  MR  Zbl
  8. K. L. Duggal, Constant scalar curvature and warped product globally null manifolds, J. Geom. Phys. 43 no. 4 (2002), 327–340.  DOI  MR  Zbl
  9. K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Math. Appl. 364, Kluwer, Dordrecht, 1996.  DOI  MR  Zbl
  10. S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monogr. Math. Phys. 1, Cambridge University Press, London-New York, 1973.  DOI  MR  Zbl
  11. S. Kumar, Geometry of warped product lightlike submanifolds of indefinite nearly Kaehler manifolds, J. Geom. 109 no. 1 (2018), article no. 21, 18 pp.  DOI  MR  Zbl
  12. S. Kumar, Warped product semi-transversal lightlike submanifolds of indefinite nearly Kaehler manifolds, Differ. Geom. Dyn. Syst. 20 (2018), 106–118.  MR  Zbl
  13. S. Kumar, A note on $SCR$-lightlike warped product submanifolds of indefinite Kaehler manifolds, Differ. Geom. Dyn. Syst. 22 (2020), 155–170.  MR  Zbl
  14. S. Kumar, Geometry of semi-transversal lightlike warped product submanifolds of indefinite Kaehler manifolds, Asian-Eur. J. Math. 14 no. 1 (2021), article no. 2150001, 12 pp.  DOI  MR  Zbl
  15. B. O'Neill, Semi-Riemannian geometry, Pure Appl. Math. 103, Academic Press, New York, 1983.  MR  Zbl
  16. N. Papaghiuc, Semi-slant submanifolds of a Kaehlerian manifold, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 40 no. 1 (1994), 55–61.  MR  Zbl
  17. B. Ṣahin, Warped product lightlike submanifolds, Sarajevo J. Math. 1(14) no. 2 (2005), 251–260.  DOI  MR  Zbl
  18. B. Sahin, Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata 117 (2006), 195–202.  DOI  MR  Zbl
  19. B. Sahin, Warped product submanifolds of Kaehler manifolds with a slant factor, Ann. Polon. Math. 95 no. 3 (2009), 207–226.  DOI  MR  Zbl
  20. S. S. Shukla and A. Yadav, Semi-slant lightlike submanifolds of indefinite Kaehler manifolds, Rev. Un. Mat. Argentina 56 no. 2 (2015), 21–37.  MR  Zbl