The main objective of this paper is to achieve the Chen-Ricci inequality for biwarped product submanifolds isometrically immersed in a complex space form in the expressions of the squared norm of mean curvature vector and warping functions.The equality cases are likewise discussed. In particular, we also derive Chen-Ricci inequality for CR-warped product submanifolds and point wise semi slant warped product submanifolds.
Citation: Amira A. Ishan, Meraj Ali Khan. Chen-Ricci inequality for biwarped product submanifolds in complex space forms[J]. AIMS Mathematics, 2021, 6(5): 5256-5274. doi: 10.3934/math.2021311
The main objective of this paper is to achieve the Chen-Ricci inequality for biwarped product submanifolds isometrically immersed in a complex space form in the expressions of the squared norm of mean curvature vector and warping functions.The equality cases are likewise discussed. In particular, we also derive Chen-Ricci inequality for CR-warped product submanifolds and point wise semi slant warped product submanifolds.
| [1] | R. L. Bishop, B. O'Neil, Manifolds of negative curvature, T. Am. Math. Soc., 145 (1969), 1-9. |
| [2] | J. K. Beem, P. Ehrlich, T. G. Powell, Warped product manifolds in relativity, In: Th. M Rassias, G. M. Rassias, Editor, Selected Studies: Physics-Astrophysics, Mathematics, History of Science, Amsterdam: North-Holland, 1982, 41-56. |
| [3] | S. W. Hawkings, G. F. R. Ellis, The large scale structure of space-time, Cambridge: Cambridge Univ. Press, 1973. |
| [4] | B. O'Neill, Semi-Riemannian geometry with application to relativity, Academic Press, 1983. |
| [5] | B. Y. Chen, CR-submanifolds of a Kaehler manifold I, J. Differ. Geom., 16 (1981), 305-323. |
| [6] | B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh Math., 133 (2001), 177-195. |
| [7] | B. Y. Chen, A survey on geometry of warped product submanifolds, arXiv: 1307.0236. |
| [8] | B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasg. Math. J., 41 (1999), 33-41. |
| [9] | K. Arslan, R. Ezentas, I. Mihai, C. Ozgur, Certain inequalities for submanifolds in $(k, \mu)-$contact space form, B. Aust. Math. Soc., 64 (2001), 201-212. |
| [10] | D. Cioroboiu, B. Y. Chen, Inequalities for semi-slant submanifolds in Sasakian space forms, IJMMS, 27 (2003), 1731-1738. |
| [11] | A. Mihai, C. Ozgur, Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwan. J. Math., 14 (2010), 1465-1477. |
| [12] | D. W. Yoon, Inequality for Ricci curvature of slant submanifolds in cosymplectic space forms, Turk. J. Math., 30 (2006), 43-56. |
| [13] | M. Aquib, J. W. Lee, G. E. Vilcu, W. Yoon, Classification of Casorati ideal Lagrangian submanifolds in complex space forms, Differ. Geom. Appl., 63 (2019), 30-49. |
| [14] | M. M. Tripathi, Improved Chen-Ricci inequality for curvature-like tensors and its applications, Differ. Geom. Appl., 29 (2011), 685-698. |
| [15] | A. Mustafa, S. Uddin, F. R. Al-Solamy, Chen-Ricci inequality for warped products in Kenmotsu space forms and its applications, RACSAM, 113 (2019), 1-18. |
| [16] | B. Y. Chen, F. Dillen, Optimal inequality for multiply warped product submanifolds, IEJG, 1 (2008), 1-11 |
| [17] | J. P. Baker, Twice warped products, M. Sc. Thesis, University of Missouri-Columbia, Columbia, 1997. |
| [18] | H. M. Tastan, Biwarped product submanifolds of a Kaehler manifold, Filomat, 32 (2018), 2349-2365. |
| [19] | M. A. Khan, K. Khan, Biwarped product submanifolds of complex space forms, Int. J. Geom. Methods Mod. Phys., 16 (2019), 1950072. |
Amira A. Ishan, Meraj Ali Khan. Chen-Ricci inequality for biwarped product submanifolds in complex space forms[J]. AIMS Mathematics, 2021, 6(5): 5256-5274. doi: 10.3934/math.2021311
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