Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

  • Mehraj Ahmad Lone NIT Srinagar, India
  • Yoshio Matsuyama Department of Mathematics, Chuo University, Faculity of Sciences and Engineering, 1-13-27 Kasuga, Bunkyo-Ku, Tokyo 112-8551, Japan
  • Falleh R. Al-Solamy Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
  • Mohammad Hasan Shahid Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India.
  • Mohammed Jamali Department of Mathematics, Al-Falah University, Haryana-121004, India.
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.,

(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),

-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.

(2008), 803-824.

A. Ghosh, R. Sharma, Contact hypersurfaces of a bochner-Kaehler manifolds, Results in math., 64(2013),

-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,

(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-

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),

-240.

Published
2020-03-25
How to Cite
Lone, M. A., Matsuyama, Y., Al-Solamy, F. R., Shahid, M. H., & Jamali, M. (2020). Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds. Tamkang Journal of Mathematics, 51(1). https://doi.org/10.5556/j.tkjm.51.2020.2967
Section
Papers