Upper Bounds for Ricci Curvatures for Submanifolds in Bochner-Kaehler Manifolds
Keywords:Bochner-Kaehler manifold, Einstein manifold, Ricci curvature, slant submanifolds.
Chen established the relationship between the Ricci curvature and the squared norm of mean curvature vector for submanifolds of Riemannian space form with arbitrary codimension known as Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complex space form by using algebraic technique. In this paper, we establish the same inequalities for different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improved
Chen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.
S. Bochner, Curvature and Betti numbers II, Ann. Math. 50(1949), 77-93.
R. L. Bryant, Bochner-Kaehler metrics, J. Amer. Math. Soc. 14(2001), 623-715.
G. Calvaruso, Nulity index of BochnerKaehler manifolds, Note Mat. 29(2009), 115-122.
B. Y. Chen, A general inequality for submanifolds in complex space forms and its applications,Arch. Math.,67(1996), 519-528.
B. Y. Chen, Geometry of Slant submanifolds, Katholieke Universitiet Leuven, 1990.
B. Y. Chen, Mean curvature and shape operator of isometric immersions in real space forms, Glasgow. Math. J., 38(1996), 87-97.
B. Y. Chen, Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow. Math. J., 41(1999), 33-41.
B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. math., 60(1993), 568-578.
B. Y. Chen, Some topological obstructions to Bochner kaehler metrics and their applications, J. Differential Geom. 13(1978), 547–558.
S. Deng, An improved Chen-Ricci inequality, Int. Electronic J. Geom., 2(2009), 39-45.
S. Deng, Improved Chen-Ricci inequality for Lagrangian submanifolds in quartenion space forms, Int. Electronic J. Geom., 1(2012), 163-170.
G. Ganchev, V. Mihova, Warped product Kaehler manifolds and bochner Kaehler metrics, J. Geom. Phys. 58(2008), 803-824.
A. Ghosh, R. Sharma, Contact hypersurfaces of a bochner-Kaehler manifolds, Results in math., 64(2013), 155-163.
C. S. Houh, Totally real submanifolds in a Bochner-Kaehler manifolds, Tensor N.S, 32(1978), 293-296.
Y. Inoue, The penrose transformation on conformally Bochner-Kaehler manifolds, J. Math. Kyoto University, 47(2007), 327-357.
J. S. Kim, Y. M. Song, M. M. Tripathi, B. Y. Chen inequalities for submanifolds in generilized complex space forms, Bull. Korean Math. Soc., 40(2003), 411-423.
M. A. Lone, M. Jamali, M. H. Shahid, On some inequalities for submanifolds of Bochner-Kaehler manifolds, To Appear in Filomat.
K. Matsumoto, I. Mihai, A. Oiaga, Ricci curvature of submanifolds in complex space form, Rev. Roumaine Math. Pures Appl., 46(2001), 775-782.
K. Matsumoto, I. Mihai, Y. Tazawa, Ricci tensor of slant submanifolds in complex space form, Kodai Math. J., 26(2003), 85-94.
I. Mihai, Inequalities on the Ricci curvature, J. Math. Inequalities , 9(2015), 811-822.
I. Mihai, Ricci curvature of submanifolds in sasakian space forms, J. Austral. Math. Soc. 72(2002), 247-256.
I. Mihai, I. N. Radulescu An improved Chen-Ricci inequality for Kaehlerian slant submanifolds in complex space form, Taiwanese J. Math. , 1(2012), 761-770.
L. J. Schwachhofer, Special connections on symplectic manifolds, Rend. Cric. Mat. Palremo, 75(2005), 197-223.
M. H. Shahid, F. R. Al-Solamy, Ricci tensor of slant submanifolds in a quaternion projective space, C. R. Acad. Sci. Paris, Ser. , 349(2011), 571-373.
M. H. Shahid, S. I. Husain, CR-submanifolds of a Bochner-Kaehler manifold, Indian J. Pure. and Applied Math., 18(1987), 605-610.
A. Song, X. Liu, Some inequalities of slant submanifolds in generilized complex space form, Tamkang Math. J., 36(2005), 223-229.
M. M. Tripathi, Improved Chen-Ricci inequality for curvature like tensors and its applications, Differ. Geom. Appl. 29(2011), 685-692.
P. Verheven, L. Verstraelen, Quasiumbilical anti-invariant submanifolds, Riv. Mat. Univ. Parma, 4(1983), 233-240.
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