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Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold

Year 2021, Volume: 50 Issue: 3, 634 - 646, 07.06.2021
https://doi.org/10.15672/hujms.585465

Abstract

The purpose of this paper is to provide the complete classifications of the second fundamental form inequality for a warped product pointwise semi-slant submanifold in a Kaehler manifold which was obtained by Şahin [Th. 5.2, Warped product pointwise semi-slant submanifold of Kaehler manifold, Port. Math. 2013] in terms of intrinsic and extrinsic invariants. In this paper, we give some conditions that are not addressed or considered in the cited paper mentioned above. Finally, we provide necessary and sufficient conditions that satisfy inequalities' equalities.

Supporting Institution

King Khalid University

Project Number

R.G.P.2/57/40

Thanks

The authors extend their appreciation to the deanship of scientific research at King Khalid University for funding this work through research groups program.

References

  • [1] A. Ali, and P. Laurian-Ioan, Geometry of warped product immersion of Kenmotsu space forms and its application to slant immersions, J. Geom. Phy. 114, 276-290, 2017.
  • [2] A. Ali and P. Laurian-Ioan, Geometric classification of warped products isometrically immersed into Sasakian space forms, Math. Nachr. 292, 234–251, 2019.
  • [3] A. Ali, and C. Özel, Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications, Int. J. Geom. Methods Mod. Phys. 14, 1750042, 2017.
  • [4] A. Ali, S. Uddin, and W.A.M. Othman, Geometry of warped product pointwise semislant submanifolds of Kaehler manifolds, Filomat 32 (12), 3771-3788, 2017.
  • [5] R.L. Bishop, and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145, 1-9, 1969.
  • [6] O. Calin and D.C. Chang, Geometric mechanics on Riemannian manifolds: applications to partial differential equations, Springer Science & Business Media, 2006.
  • [7] B.-Y. Chen, Geometry of warped product CR-submanifold in Kaehler manifolds I, Monatsh. Math. 133, 177-195, 2001.
  • [8] B.-Y. Chen, Another general inequality for warped product CR-warped product submanifold in complex space forms, Hokkaido Math. J. 32, 415-444, 2003.
  • [9] B.-Y. Chen, and O. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turkish J. Math. 36, 630-640, 2012.
  • [10] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen 53, 217-223, 1998.
  • [11] A. Mustafa, Geometry of warped product submanifolds of Riemannian manifolds, PhD thesis, University of Malaya, 2016.
  • [12] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. 63, 2063, 1956.
  • [13] N. Papaghiuc, Semi-slant submanifold of Kaehler manifold, An. St. Univ. Al. I. Cuza. Iasi. 40, 55-61, 1994.
  • [14] B. Şahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata 117, 95202, 2006.
  • [15] B. Şahin, Warped product pointwise semi-slant submanifold of Kaehler manifold, Port. Math. 70, 251-268, 2013.
  • [16] M.M. Tripathi, Improved Chen-Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29, 685-698, 2011.
  • [17] M. M. Tripathi, C-totally real warped product submanifolds, An. St. Univ. Al. I. Cuza. Iasi 58 (2), 417-436, 2012.
  • [18] B. Ünal, Multiply Warped Products, J. Geom. Phy. 34 (3-4), 287-301, 2000.
  • [19] K. Yano and M. Kon, Structures on manifolds, World Scientific, 1984.
Year 2021, Volume: 50 Issue: 3, 634 - 646, 07.06.2021
https://doi.org/10.15672/hujms.585465

Abstract

Project Number

R.G.P.2/57/40

References

  • [1] A. Ali, and P. Laurian-Ioan, Geometry of warped product immersion of Kenmotsu space forms and its application to slant immersions, J. Geom. Phy. 114, 276-290, 2017.
  • [2] A. Ali and P. Laurian-Ioan, Geometric classification of warped products isometrically immersed into Sasakian space forms, Math. Nachr. 292, 234–251, 2019.
  • [3] A. Ali, and C. Özel, Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications, Int. J. Geom. Methods Mod. Phys. 14, 1750042, 2017.
  • [4] A. Ali, S. Uddin, and W.A.M. Othman, Geometry of warped product pointwise semislant submanifolds of Kaehler manifolds, Filomat 32 (12), 3771-3788, 2017.
  • [5] R.L. Bishop, and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145, 1-9, 1969.
  • [6] O. Calin and D.C. Chang, Geometric mechanics on Riemannian manifolds: applications to partial differential equations, Springer Science & Business Media, 2006.
  • [7] B.-Y. Chen, Geometry of warped product CR-submanifold in Kaehler manifolds I, Monatsh. Math. 133, 177-195, 2001.
  • [8] B.-Y. Chen, Another general inequality for warped product CR-warped product submanifold in complex space forms, Hokkaido Math. J. 32, 415-444, 2003.
  • [9] B.-Y. Chen, and O. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turkish J. Math. 36, 630-640, 2012.
  • [10] F. Etayo, On quasi-slant submanifolds of an almost Hermitian manifold, Publ. Math. Debrecen 53, 217-223, 1998.
  • [11] A. Mustafa, Geometry of warped product submanifolds of Riemannian manifolds, PhD thesis, University of Malaya, 2016.
  • [12] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. 63, 2063, 1956.
  • [13] N. Papaghiuc, Semi-slant submanifold of Kaehler manifold, An. St. Univ. Al. I. Cuza. Iasi. 40, 55-61, 1994.
  • [14] B. Şahin, Non-existence of warped product semi-slant submanifolds of Kaehler manifolds, Geom. Dedicata 117, 95202, 2006.
  • [15] B. Şahin, Warped product pointwise semi-slant submanifold of Kaehler manifold, Port. Math. 70, 251-268, 2013.
  • [16] M.M. Tripathi, Improved Chen-Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29, 685-698, 2011.
  • [17] M. M. Tripathi, C-totally real warped product submanifolds, An. St. Univ. Al. I. Cuza. Iasi 58 (2), 417-436, 2012.
  • [18] B. Ünal, Multiply Warped Products, J. Geom. Phy. 34 (3-4), 287-301, 2000.
  • [19] K. Yano and M. Kon, Structures on manifolds, World Scientific, 1984.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ali H. Alkhaldi This is me

Akram Ali 0000-0002-6053-3031

Mehmet Akif Akyol 0000-0003-2334-6955

Project Number R.G.P.2/57/40
Publication Date June 7, 2021
Published in Issue Year 2021 Volume: 50 Issue: 3

Cite

APA Alkhaldi, A. H., Ali, A., & Akyol, M. A. (2021). Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold. Hacettepe Journal of Mathematics and Statistics, 50(3), 634-646. https://doi.org/10.15672/hujms.585465
AMA Alkhaldi AH, Ali A, Akyol MA. Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold. Hacettepe Journal of Mathematics and Statistics. June 2021;50(3):634-646. doi:10.15672/hujms.585465
Chicago Alkhaldi, Ali H., Akram Ali, and Mehmet Akif Akyol. “Some Remarks on a Family of Warped Product Pointwise Semi-Slant Submanifolds of Kaehler Manifold”. Hacettepe Journal of Mathematics and Statistics 50, no. 3 (June 2021): 634-46. https://doi.org/10.15672/hujms.585465.
EndNote Alkhaldi AH, Ali A, Akyol MA (June 1, 2021) Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold. Hacettepe Journal of Mathematics and Statistics 50 3 634–646.
IEEE A. H. Alkhaldi, A. Ali, and M. A. Akyol, “Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 634–646, 2021, doi: 10.15672/hujms.585465.
ISNAD Alkhaldi, Ali H. et al. “Some Remarks on a Family of Warped Product Pointwise Semi-Slant Submanifolds of Kaehler Manifold”. Hacettepe Journal of Mathematics and Statistics 50/3 (June 2021), 634-646. https://doi.org/10.15672/hujms.585465.
JAMA Alkhaldi AH, Ali A, Akyol MA. Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold. Hacettepe Journal of Mathematics and Statistics. 2021;50:634–646.
MLA Alkhaldi, Ali H. et al. “Some Remarks on a Family of Warped Product Pointwise Semi-Slant Submanifolds of Kaehler Manifold”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, 2021, pp. 634-46, doi:10.15672/hujms.585465.
Vancouver Alkhaldi AH, Ali A, Akyol MA. Some remarks on a family of warped product pointwise semi-slant submanifolds of Kaehler manifold. Hacettepe Journal of Mathematics and Statistics. 2021;50(3):634-46.