Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure
DOI:
https://doi.org/10.5269/bspm.42780Abstract
This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases. \\
References
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2. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh Math., 133(2001), 177-195. https://doi.org/10.1007/s006050170019
3. B. Y. Chen, Pseudo-Riemannian Geometry, δ−invariants and Applications, World Scientific Publishing Company, Singapore, 2011. https://doi.org/10.1142/8003
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5. D. E. Blair, D. K. Showers, Almost contact manifolds with Killing structure tensors II, Journal of Differential Geometry, 9(1974), 577--582. https://doi.org/10.4310/jdg/1214432556
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8. F. R. Al-Solamy, Meraj Ali Khan, Semi-invariant warped product submanifolds of almost contact manifolds, Journal of inequalities and applications, https://doi.org/10.1186/1029-242X-2012-127
9. G. D. Ludden, Submanifolds of cosymplectic manifolds, Journal of Differential Geometry, 4(1970), 237--244. https://doi.org/10.4310/jdg/1214429387
10. I. Hasegawa, I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geometriae Dedicata, 102 (2003), 143-150. https://doi.org/10.1023/B:GEOM.0000006582.29685.22
11. I. Mihai, Contact CR-warped product submanifolds in Sasakain space forms, Geometriae Dedicata, 109(2004), 165-173. https://doi.org/10.1007/s10711-004-5459-z
12. K. Arslan, R. Ezentas, I. Mihai, C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, Journal of Korean Math. Soc., 42(5)2005, 1101-1110. https://doi.org/10.4134/JKMS.2005.42.5.1101
13. Khalid Ali Khan, Viqar Azam Khan and Siraj-Uddin, Warped product submanifolds of cosymplectic manifolds, Balkan Journal of Geometry and its Applications, 13, No.1 (2008), 55-65.
14. M. Atceken, Contact CR-warped product submanifolds in cosymplectic space forms, Collectanea Mathematica, 62 (1)(2011), 17-26. https://doi.org/10.1007/s13348-010-0002-z
15. Meraj Ali Khan, S. Uddin, R. Sachdevadeva, Semi-invariant warped product submanifolds of cosymplectic manifolds, Journal of inequalities and applications, https://doi.org/10.1186/1029-242X-2012-19
16. P. Alegre, D. E. Blair, A. Carriazo, Generalized Sasakian space forms, Israel J. Math., 141 (2004),157-183. https://doi.org/10.1007/BF02772217
17. R. L. Bishop, B. O'Neill, Manifolds of Negative curvature, Trans Amer. Math. Soc., 145(1965), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
18. S. Uddin, S. H. Kon, Meraj Ali Khan, K. Singh, Warped product semi-invariant submanifolds of nearly cosymplectic manifolds, Math. Prob. Eng., 2011, 230374. https://doi.org/10.1155/2011/230374
19. Reem Al-Ghefari, Falleh R. Al-Solamy and Mohammed H. Shahid, Contact CR-warped product submanifolds in generalized Sasakian Space Forms, 11 (1) (2006), 1-10
20. S. T. Hong, Warped products and black holes, Nuovo Cim J. B., 120(2005), 1227-1234.
2. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh Math., 133(2001), 177-195. https://doi.org/10.1007/s006050170019
3. B. Y. Chen, Pseudo-Riemannian Geometry, δ−invariants and Applications, World Scientific Publishing Company, Singapore, 2011. https://doi.org/10.1142/8003
4. D. E. Blair, Almost contact manifolds with killing structure tensors I., Pac. J. Math. 39(1971), 285-292. https://doi.org/10.2140/pjm.1971.39.285
5. D. E. Blair, D. K. Showers, Almost contact manifolds with Killing structure tensors II, Journal of Differential Geometry, 9(1974), 577--582. https://doi.org/10.4310/jdg/1214432556
6. D. E. Blair, K. Yano, Affine almost contact manifolds and f-manifolds with affine killing structure tensors, Kodai math. Semin. Rep. 23(1971), 473-479. https://doi.org/10.2996/kmj/1138846415
7. D. E. Blair, Contact manifolds in Riemannian Geometry Lecture Notes in Math. 509, Berlin: Springer-Verlag, 1976. https://doi.org/10.1007/BFb0079307
8. F. R. Al-Solamy, Meraj Ali Khan, Semi-invariant warped product submanifolds of almost contact manifolds, Journal of inequalities and applications, https://doi.org/10.1186/1029-242X-2012-127
9. G. D. Ludden, Submanifolds of cosymplectic manifolds, Journal of Differential Geometry, 4(1970), 237--244. https://doi.org/10.4310/jdg/1214429387
10. I. Hasegawa, I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geometriae Dedicata, 102 (2003), 143-150. https://doi.org/10.1023/B:GEOM.0000006582.29685.22
11. I. Mihai, Contact CR-warped product submanifolds in Sasakain space forms, Geometriae Dedicata, 109(2004), 165-173. https://doi.org/10.1007/s10711-004-5459-z
12. K. Arslan, R. Ezentas, I. Mihai, C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, Journal of Korean Math. Soc., 42(5)2005, 1101-1110. https://doi.org/10.4134/JKMS.2005.42.5.1101
13. Khalid Ali Khan, Viqar Azam Khan and Siraj-Uddin, Warped product submanifolds of cosymplectic manifolds, Balkan Journal of Geometry and its Applications, 13, No.1 (2008), 55-65.
14. M. Atceken, Contact CR-warped product submanifolds in cosymplectic space forms, Collectanea Mathematica, 62 (1)(2011), 17-26. https://doi.org/10.1007/s13348-010-0002-z
15. Meraj Ali Khan, S. Uddin, R. Sachdevadeva, Semi-invariant warped product submanifolds of cosymplectic manifolds, Journal of inequalities and applications, https://doi.org/10.1186/1029-242X-2012-19
16. P. Alegre, D. E. Blair, A. Carriazo, Generalized Sasakian space forms, Israel J. Math., 141 (2004),157-183. https://doi.org/10.1007/BF02772217
17. R. L. Bishop, B. O'Neill, Manifolds of Negative curvature, Trans Amer. Math. Soc., 145(1965), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
18. S. Uddin, S. H. Kon, Meraj Ali Khan, K. Singh, Warped product semi-invariant submanifolds of nearly cosymplectic manifolds, Math. Prob. Eng., 2011, 230374. https://doi.org/10.1155/2011/230374
19. Reem Al-Ghefari, Falleh R. Al-Solamy and Mohammed H. Shahid, Contact CR-warped product submanifolds in generalized Sasakian Space Forms, 11 (1) (2006), 1-10
20. S. T. Hong, Warped products and black holes, Nuovo Cim J. B., 120(2005), 1227-1234.
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2021-12-16
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