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DOI: 10.46698/t2068-3621-5954-b
On Generalized Kenmotsu Manifolds as Hypersurfaces of Vaisman-Gray Manifolds
Abass, M. Y.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 1.
Abstract: In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's tensors in the Vaisman-Gray class are equal to zero. On the other hand, the minimality of totally umbilical, totally geodesic hypersurfaces of Vaisman-Gray manifolds with generalized Kenmotsu structures are investigated. In addition, we deduced that the hypersurface of Vaisman-Gray manifold that have generalized Kenmotsu structure is totally geodesic if and only if it is totally umbilical and some components of Lee form are constants.
Keywords: almost contact metric manifolds, nearly Kenmotsu manifolds, Vaisman-Gray manifolds, hypersurfaces of almost Hermitian manifolds.
For citation: Abass, M. Y. On Generalized Kenmotsu Manifolds as Hypersurfaces of Vaisman-Gray Manifolds, Vladikavkaz Math. J., 2024, vol. 26, no. 1, pp. 5-12.
DOI 10.46698/t2068-3621-5954-b
1. Kenmotsu, K. A Class of Almost Contact Riemannian Manifolds,
Tohoku Mathematical Journal, 1972, vol. 24, no. 1, pp. 93-103.
DOI: 10.2748/tmj/1178241594.
2. Umnova, S. V.Geometry of Kenmotsu Manifolds and Their Generalizations,
Phd Thesis, Moscow State Pedagogical University, 2002 (in Russian).
3. Erken, I. K., Dacko, P. and Murathan, C.
On the Existence of Proper Nearly Kenmotsu Manifolds,
Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4497-4507.
DOI: 10.1007/s00009-016-0758-9.
4. Heidari, N., Kashani, N. H. P. and Najafi, B. Nearly Kahler and Nearly Kenmotsu Manifolds,
Turkish Journal of Mathematics, 2018, vol. 42, no. 3, pp. 967-976.
DOI: 10.3906/mat-1703-11.
5. Abu-Saleem, A. and Rustanov, A. R. Curvature Identities Special Generalized Manifolds Kenmotsu Second Kind,
Malaysian Journal of Mathematical Sciences, 2015, vol. 9, no. 2, pp. 187-207.
6. Abood, H. M. and Abass, M. Y. A Study of New Class of Almost Contact Metric Manifolds of Kenmotsu Type,
Tamkang Journal of Mathematics, 2021, vol. 52, no. 2, pp. 253-266.
DOI: 10.5556/j.tkjm.52.2021.3276.
7. Banaru, M. B. and Kirichenko, V. F. Almost Contact Metric Structures on the Hypersurface of
Almost Hermitian Manifolds, Journal of Mathematical Sciences, 2015, vol. 207, no. 4,
pp. 513-537. DOI: 10.1007/s10958-015-2382-9.
8. Stepanova, L. V., Banaru, M. B. and Banaru, G. A. On Geometry of \(QS\)-Hypersurfaces of Kahlerian Manifolds,
Siberian Elektronic Mathematical Reports, 2018, vol. 15, pp. 815-822 (in Russian). DOI: 10.17377/semi.2018.15.068.
9. Banaru, M. B. and Banaru, G. A. On Hypersurfaces with Kirichenko-Uskorev Structure
in Kahlerian Manifolds, Siberian Elektronic Mathematical Reports, 2020, vol. 17,
pp. 1715-1721 (in Russian). DOI: 10.33048/semi.2020.17.116.
10. Kirichenko, V. F. and Uskorev, I. V. Invariants of Conformal Transformations of Almost Contact Metric Structures, Mathematical Notes, 2008, vol. 84, no. 6, pp. 783-794. DOI: 10.1134/S0001434608110229.
11. Banaru, M. B., Banaru, G. A. and Melekhina, T. L. A Note on Almost Contact Metric 2- and 3-Hypersurfaces in \(W_{4}\)-Manifolds, Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, 2019, vol. 89, no. 1, pp. 103-108.
12. Alegre, P., Barrera, J. and Carriazo, A. A New Class of Slant Submanifolds in Generalized Sasakian Space Forms,
Mediterranean Journal of Mathematics, 2020, vol. 17, no. 3, article 76. DOI: 10.1007/s00009-020-01511-9.
13. Ignatochkina, L. A. Local Structure of Vaisman-Gray Manifolds,
Journal of Mathematical Sciences, 2016, vol. 217, no. 5, pp. 595-606.
DOI: 10.1007/s10958-016-2992-x.
14. Abu-Saleem, A., Kochetkov, I. D. and Rustanov, A. R. Two Subclasses of Generalized Kenmotsu Manifolds,
IOP Conference Series: Materials Science and Engineering, 2020, vol. 918, article 012062.
DOI: 10.1088/1757-899X/918/1/012062.
15. Kirichenko, V. F. Differential-Geometric Structures on Manifolds,
2nd edition, Odessa, Pechatnyy Dom, 2013 (in Russian).
16. Rustanov, A. R., Kazakova, O. N. and Kharitonova, S. V. Contact Analogs of Gray's Identity for \(NC_{10}\)-Manifolds, Siberian Elektronic Mathematical Reports, 2018, vol. 15, pp. 823-828 (in Russian). DOI: 10.17377/semi.2018.15.069.
17. Abu-Saleem, A., Kochetkov, I. D. and Rustanov, A. R. Curvature Identities for Generalized Kenmotsu Manifolds, E3S Web of Conferences, 2021, vol. 244, article 09005. DOI: 10.1051/e3sconf/202124409005.
18. Ignatochkina, L. A. Vaisman-Gray Manifolds with \(J\)-Invariant Conformal Curvature Tensor, Sbornik: Mathematics, 2003, vol. 194, no. 2, pp. 225-235. DOI: 10.1070/sm2003v194n02abeh000713.
19. Chen, B.-Y. Differential Geometry of Warped Product Manifolds and Submanifolds, Singapore, World Scientific, 2017.
20. Abu-Saleem, A. and Banaru, M. B. Two Theorems on Kenmotsu Hypersurfaces in a \(W_{3}\)-Manifold,
Studia Universitatis Babes-Bolyai, Mathematica, 2005, vol. 50, no. 3, pp. 3-11.
21. Balkana, Y. S. and Alkhaldi, A. H. Chen's Type Inequality for Warped Product Pseudo-Slant Submanifolds of Kenmotsu \(f\)-Manifolds, Filomat, 2019, vol. 33, no. 11, pp. 3521-3536. DOI: 10.2298/FIL1911521B.
