Abstract
In the present paper, we prove that there does not exist any \(\mathcal {P}\mathcal {R}\)-semi-slant warped product submanifolds in paracosymplectic manifolds. In addition, by presenting a non-trivial example we find that there is no proper \(\mathcal {P}\mathcal {R}\)-semi-slant warped product submanifold other than \(\mathcal {P}\mathcal {R}\)-semi-invariant warped products.
Article PDF
Similar content being viewed by others
References
Alegre, P.; Carriazo, A.: Slant submanifolds of para-Hermitian manifolds. Mediterr. J. Math. 14, 214 (2017). https://doi.org/10.1007/s00009-017-1018-3
Aydin, G.; Cöken, A.C.: Slant submanifolds of semi-Riemannian manifolds. Int. J. Geom. Methods Mod. Phys. Math. 10(10), 1350058 (2013). (13 pages)
Alegre, P.: Slant submanifolds of Lorentzian Sasakian and para-Sasakian manifolds. Taiwan. J. Math. 17(3), 897–910 (2013)
Bejancu, A.; Papaghiuc, N.: Semi-invariant submanifolds of a Sasakian space form. Colloq. Math. 48(2), 229–240 (1984)
Bishop, R.L.; O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 01–49 (1969)
Cabrerizo, J.L.; Carriazo, A.; Fernández, L.M.; Fernández, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedic. 78, 183–199 (1999)
Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133(3), 177–195 (2001)
Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. II. Ibid. 134(2), 103–119 (2001)
Chen, B.-Y.: CR-submanifolds in Kaehler manifolds. I. J. Differ. Geom. 16, 305–322 (1981)
Chen, B.-Y.: CR-submanifolds in Kaehler manifolds II. Ibid 16, 493–509 (1981)
Chen, B.-Y.: Geometry of Submanifolds. Marcel Dekker Inc, New York (1973)
Chen, B.-Y.: Pseudo-Riemannian Geometry, \(\delta \)-Invariants and Applications. Word Scientific, Hackensack (2017)
Chen, B.-Y.; Munteanu, M.I.: Geometry of \(\cal{P}\cal{R}\)-warped products in para-Kähler manifolds. Taiwan. J. Math. 16(4), 1293–1327 (2012)
Chen, B.-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6(2), 01–43 (2013)
Chen, B.-Y.; Uddin, S.: Warped product pointwise bi-slant submanifolds of Kaehler manifolds. Publ. Math. Debr. 92(1–2), 183–199 (2018)
Chen, B.-Y.: Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version. arXiv:1806.11102v1 [math.DG].
Dacko, P.: On almost para-cosymplectic manifolds. Tsukuba J. Math. 28(1), 193–213 (2004)
Etayo, F.; Fioravanti, M.; Trias, U.R.: On the submanifolds of an almost para-Hermitian manifold. Acta. Math. Hung. 85(4), 277–286 (1999)
Hasegawa, I.; Mihai, I.: Contact CR-warped product submanifolds in Sasakian manifolds. Geom. Dedic. 102, 143–150 (2003)
Hong, S.T.: Warped products and black holes. Nuovo Cim. Soc. Ital. Fis. B. 120, 1227–1234 (2005)
Katanaev, M.O.; Klösch, T.; Kummer, W.: Global properties of warped solutions in general relativity. Ann. Phys. 276, 191–222 (1999)
Kaneyuki, S.; Williams, F.L.: Almost paracontact and parahodge structures on manifolds. Nagoya Math. J. 99, 173–187 (1985)
Khan, K.A.; Khan, V.A.; Uddin, S.: Warped product submanifolds of cosymplectic manifolds. Balkan J. Geom. Appl. 13, 55–65 (2008)
Lotta, A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Roum. 39, 183–198 (1996)
Munteanu, M.I.: Warped product contact CR submanifolds in Sasakian space forms. Publ. Math. Debr. 66, 75–120 (2005)
Montano, B.C.: Bi-legendrian structures and paracontact geometry. Int. J. Geom. Methods Mod. Phys. 6(3), 487–504 (2009)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New york (1983)
Sahin, B.: Non-existence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedic. 117(1), 195–202 (2006)
Srivastava, K.; Srivastava, S.K.: On a class of \(\alpha \)-para Kenmotsu manifolds. Mediterr. J. Math. 13(1), 391–399 (2016)
Srivastava, S.K.; Sharma, A.: Geometry of \(\cal{P}\cal{R}\)-semi-invariant warped product submanifolds in paracosymplectic manifold. J. Geom. 108(1), 61–74 (2017)
Srivastava, S.K., Sharma, A., Tiwari, S.K.: On \(\cal{P}\cal{R}\)-pseudo-slant warped product submanifolds in a nearly paracosymplectic manifold. Ann. St. Al. I. Cuza Univ. Iasi. Romania (2017). (Accepted)
Srivastava, S.K.; Sharma, A.: Pointwise pseudo-slant warped product submanifolds in a Kähler manifold. Mediterr. J. Math. 14, 20 (2017). https://doi.org/10.1007/s00009-016-0832-3
Uddin, S.: Warped product CR-submanifolds of LP-cosymplectic manifolds. Filomat. 24(1), 87–95 (2010)
Uddin, S.; Khan, V.A.; Khan, K.A.: A note on warped product submanifolds of cosymplectic manifolds. Filomat 24(3), 95–102 (2010)
Uddin, S.; Khan, K.A.: Warped product CR-submanifolds of cosymplectic manifolds. Ricerche di Matematica 60, 143–149 (2011)
Uddin, S.; Alqahtani, L.S.: Chen type inequality for warped product immersions in cosymplectic space forms. J. Nonlinear Sci. Appl. 9, 2914–2921 (2016)
Uddin, S.; Al-Solamy, F.R.: Another proof of derived inequality for warped product semi-invariant submanifolds of cosymplectic manifolds. J. Math. Anal. 7(4), 93–97 (2016)
Uddin, S.; Chen, B.-Y.; Al-Solamy, F.R.: Warped product bi-slant immersions in Kaehler manifolds. Mediterr. J. Math. 14, 95 (2017). https://doi.org/10.1007/s00009-017-0896-8
Uddin, S.; Al-Solamy, F.R.: Warped product pseudo-slant immersions in Sasakian manifolds. Publ. Math. Debr. 91(3–4), 331–348 (2017)
Uddin, S.: Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds. Bull. Math. Sci. 8(3), 435–451 (2018). https://doi.org/10.1007/s13373-017-0106-9
Uddin, S.; Naghi, M.F.; Al-Solamy, F.R.: Another class of warped product submanifolds of Kenmotsu manifolds. RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 112(4), 1141–1155 (2018). https://doi.org/10.1007/s13398-017-0415-6
Yüksel, P.S.; KıIıç, E.; Keleş, S.: Warped product submanifolds of Lorentzian paracosymplectic manifolds. Arab. J. Math. 1(3), 377–393 (2012)
Zamkovoy, S.: Canonical connections on paracontact manifolds. Ann. Glob. Anal. Geom. 36, 37–60 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sharma, A., Uddin, S. & Srivastava, S.K. Nonexistence of \(\mathcal {P}\mathcal {R}\)-semi-slant warped product submanifolds in paracosymplectic manifolds. Arab. J. Math. 9, 181–190 (2020). https://doi.org/10.1007/s40065-018-0234-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-018-0234-4