Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Non Existence of 𝒫ℛ-semi-slant Warped Product Submanifolds in a Para-Kähler Manifold

  • Sharma, Anil (Chitkara University Institute of Engineering and Technology, Chitkara University)
  • Received : 2018.12.26
  • Accepted : 2020.01.16
  • Published : 2020.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove that there are no non-trivial 𝒫ℛ-semi-slant warped product submanifolds with proper slant coefficients in para-Kähler manifolds ${\bar{M}}$. We also present a numerical example that illustrates the existence of a 𝒫ℛ-warped product submanifold in ${\bar{M}}$.

Keywords

References

  1. P. Alegre and A. Carriazo, Slant submanifolds of para-Hermitian manifolds, Mediterr. J. Math., 14(5)(2017) Art. 214, 14 pp.
  2. G. Aydin and A. C. Coken, Slant submanifolds of semi-Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(10)(2013), 1350058, 13 pp. https://doi.org/10.1142/S0219887813500588
  3. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145(1969), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
  4. B. Y. Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973.
  5. B. Y. Chen, CR-submanifolds of a Kaehler manifold I, J. Differential Geometry, 16(1981), 305-322 https://doi.org/10.4310/jdg/1214436106
  6. B. Y. Chen, CR-submanifolds of a Kaehler manifold II, J. Differential Geometry, 16(1981), 493-509. https://doi.org/10.4310/jdg/1214436226
  7. B. Y. Chen, Slant immersion, Bull. Austral. Math. Soc., 41(1990), 135-147. https://doi.org/10.1017/S0004972700017925
  8. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math., 133(3)(2001), 177-195 https://doi.org/10.1007/s006050170019
  9. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Monatsh. Math., 134(2)(2001), 103-119. https://doi.org/10.1007/s006050170002
  10. B. Y. Chen, Pseudo-Riemannian geometry ${\delta}$-invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
  11. B. Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  12. B. Y. Chen, Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version, (2018), preprint, arXiv:1806.11102v1.
  13. B. Y. Chen and M. I. Munteanu, Geometry of PR-warped products in para-Kahler manifolds, Taiwanese J. Math., 16(4)(2012), 1293-1327. https://doi.org/10.11650/twjm/1500406736
  14. J. Davidov, G. Grantcharov and O. Mushkarov, Geometry of neutral metric in dimension four, (2008), preprint, arXiv:0804.2132v1.
  15. M. Dunajsky and S. West, Anti-self-dual conformal structures in neutral signature, Recent developments in pseudo-Riemannian Geometry, 113-148, ESI Lect. Math. Phys., Eur. Math. Soc., Zurich, 2008.
  16. F. Etayo, M. Fioravanti and U. R. Trias, On the submanifolds of an almost parahermitian manifold, Acta. Math. Hungar., 85(4)(1999), 277-286. https://doi.org/10.1023/A:1006779919681
  17. I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata, 102(2003), 143-150. https://doi.org/10.1023/B:GEOM.0000006582.29685.22
  18. M. O. Katanaev, T. Klosch and W. Kummer, Global Properties of warped solutions in general relativity, Ann. Physics, 276(1999), 191-222. https://doi.org/10.1006/aphy.1999.5923
  19. P. V. Kreitler, Trends in black hole research, Nova Sci. Publishers, New York, 2005.
  20. P. Libermann, Sur le probleme dequivalence de certaines structures infnitesimales, Ann. Mat. Pura Appl, 36(1954), 27-120. https://doi.org/10.1007/BF02412833
  21. M. I. Munteanu, Warped product contact CR-submanifolds of Sasakian space forms, Publ. Math. Debrecen, 66(2005), 75-120.
  22. B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
  23. H. Ooguri and C. Vafa, Self-duality and N = 2 string magic, Modern Phys. Lett. A, 5(18)(1990), 1389-1398. https://doi.org/10.1142/S021773239000158X
  24. J. Petean, Indefinite Kahler-Einstein metrics on compact complex surfaces, Commun. Math. Phys., 189(1997), 227-235. https://doi.org/10.1007/s002200050197
  25. B. Sahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata, 117(2006), 195-202. https://doi.org/10.1007/s10711-005-9023-2
  26. A. Sharma and S. K. Srivastava, Non-existence of PR-pseudo slant warped product submanifolds in paracosymplectic manifolds, Arab. J. Math. (Springer), 9(1)(2020), 181--190. https://doi.org/10.1007/s40065-018-0234-4
  27. S. K. Srivastava and A. Sharma, Pointwise pseudo-slant warped product submanifolds in a Kahler manifold, Mediterr. J. Math., 14(2017), Art. 20, 18 pp.