# Upper Bounds for Ricci Curvatures for Submanifolds in Bochner-Kaehler Manifolds

## DOI:

https://doi.org/10.5556/j.tkjm.51.2020.2967## Keywords:

Bochner-Kaehler manifold, Einstein manifold, Ricci curvature, slant submanifolds.## Abstract

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.

## References

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.

### Additional Files

## Published

## How to Cite

*Tamkang Journal of Mathematics*,

*51*(1), 53–67. https://doi.org/10.5556/j.tkjm.51.2020.2967

## Issue

## Section

## License

Copyright (c) 2020 Tamkang Journal of Mathematics

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.