Browse > Article
http://dx.doi.org/10.5666/KMJ.2020.60.1.197

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)
Publication Information
Kyungpook Mathematical Journal / v.60, no.1, 2020 , pp. 197-210 More about this Journal
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
slant submanifold; warped product; para-$K{\ddot{a}}hler$ manifold; pseudo-Riemannian manifold;
Citations & Related Records
연도 인용수 순위
  • Reference
1 P. Libermann, Sur le probleme dequivalence de certaines structures infnitesimales, Ann. Mat. Pura Appl, 36(1954), 27-120.   DOI
2 M. I. Munteanu, Warped product contact CR-submanifolds of Sasakian space forms, Publ. Math. Debrecen, 66(2005), 75-120.
3 B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
4 H. Ooguri and C. Vafa, Self-duality and N = 2 string magic, Modern Phys. Lett. A, 5(18)(1990), 1389-1398.   DOI
5 J. Petean, Indefinite Kahler-Einstein metrics on compact complex surfaces, Commun. Math. Phys., 189(1997), 227-235.   DOI
6 B. Sahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata, 117(2006), 195-202.   DOI
7 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.   DOI
8 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.
9 B. Y. Chen, CR-submanifolds of a Kaehler manifold II, J. Differential Geometry, 16(1981), 493-509.   DOI
10 P. Alegre and A. Carriazo, Slant submanifolds of para-Hermitian manifolds, Mediterr. J. Math., 14(5)(2017) Art. 214, 14 pp.
11 G. Aydin and A. C. Coken, Slant submanifolds of semi-Riemannian manifolds, Int. J. Geom. Methods Mod. Phys., 10(10)(2013), 1350058, 13 pp.   DOI
12 R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145(1969), 1-49.   DOI
13 B. Y. Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973.
14 B. Y. Chen, CR-submanifolds of a Kaehler manifold I, J. Differential Geometry, 16(1981), 305-322   DOI
15 B. Y. Chen, Slant immersion, Bull. Austral. Math. Soc., 41(1990), 135-147.   DOI
16 B. Y. Chen, Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version, (2018), preprint, arXiv:1806.11102v1.
17 B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds II, Monatsh. Math., 134(2)(2001), 103-119.   DOI
18 B. Y. Chen, Pseudo-Riemannian geometry ${\delta}$-invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
19 B. Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
20 B. Y. Chen and M. I. Munteanu, Geometry of PR-warped products in para-Kahler manifolds, Taiwanese J. Math., 16(4)(2012), 1293-1327.   DOI
21 I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata, 102(2003), 143-150.   DOI
22 J. Davidov, G. Grantcharov and O. Mushkarov, Geometry of neutral metric in dimension four, (2008), preprint, arXiv:0804.2132v1.
23 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.
24 F. Etayo, M. Fioravanti and U. R. Trias, On the submanifolds of an almost parahermitian manifold, Acta. Math. Hungar., 85(4)(1999), 277-286.   DOI
25 B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Monatsh. Math., 133(3)(2001), 177-195   DOI
26 M. O. Katanaev, T. Klosch and W. Kummer, Global Properties of warped solutions in general relativity, Ann. Physics, 276(1999), 191-222.   DOI
27 P. V. Kreitler, Trends in black hole research, Nova Sci. Publishers, New York, 2005.