# RICCI CURVATURE OF SUBMANIFOLDS OF AN S-SPACE FORM

• Published : 2009.09.30
• 57 5

#### Abstract

Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for a submanifold of an S-space form tangent to structure vector fields. Equality cases are also discussed. As applications we find corresponding results for almost semi-invariant submanifolds, $\theta$-slant submanifolds, anti-invariant submanifold and invariant submanifolds. A necessary and sufficient condition for a totally umbilical invariant submanifold of an S-space form to be Einstein is obtained. The inequalities for scalar curvature and a Riemannian invariant $\Theta_k$ of different kind of submanifolds of a S-space form $\tilde{M}(c)$ are obtained.

#### Keywords

S-space form;almost semi-invariant submanifold;$\theta$-slant submanifold;anti-invariant submanifold;Ricci curvature;k-Ricci curvature;scalar curvature;$\Theta$-invaraint

#### References

1. A. Bejancu, Geometry of CR-Submanifolds, D. Reidel Publishing Co., Dordrecht, 1986
2. D. E. Blair, Geometry of manifolds with structural group $U(n)\;{\times}\;O(s)$, J. Differential Geometry 4 (1970), 155–167
3. D. E. Blair, On a generalization of the Hopf fibration, An. Sti. Univ. “Al. I. Cuza” Ia,si Sect. I a Mat. (N.S.) 17 (1971), 171–177
4. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2002
5. D. E. Blair, G. D. Ludden, and K. Yano, Differential geometric structures on principal toroidal bundles, Trans. Amer. Math. Soc. 181 (1973), 175–184 https://doi.org/10.2307/1996627
6. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, A classification of totally fumbilical submanifolds of an S-manifold, Soochow J. Math. 18 (1992), no. 2, 211–221
7. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, On certain anti-invariant submanifolds of an S-manifold, Portugal. Math. 50 (1993), no. 1, 103–113
8. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, On normal CR-submanifolds of S-manifolds, Colloq. Math. 64 (1993), no. 2, 203–214
9. A. Carriazo, L. M. Fern´andez, and M. B. Hans-Uber, Minimal slant submanifolds of the smallest dimension in S-manifolds, Rev. Mat. Iberoamericana 21 (2005), no. 1, 47–66
10. A. Carriazo, L. M. Fernandez, and M. B. Hans-Uber, Some slant submanifolds of S-manifolds, Acta Math. Hungar. 107 (2005), no. 4, 267–285 https://doi.org/10.1007/s10474-005-0195-x
11. B.-Y. Chen, Mean curvature and shape operator of isometric immersions in real-spaceforms, Glasgow Math. J. 38 (1996), no. 1, 87–97 https://doi.org/10.1017/S001708950003130X
12. B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J. 41 (1999), no. 1, 33–41 https://doi.org/10.1017/S0017089599970271
13. B.-Y. Chen, Riemannian submanifolds, Handbook of differential geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000
14. L. M. Fernandez and M. B. Hans-Uber, New relationships involving the mean curvature of slant submanifolds in S-space-forms, J. Korean Math. Soc. 44 (2007), no. 3, 647–659 https://doi.org/10.4134/JKMS.2007.44.3.647
15. S. P. Hong and M. M. Tripathi, On Ricci curvature of submanifolds of generalized Sasakian space forms, Int. J. Pure Appl. Math. Sci. 2 (2005), no. 2, 173–201
16. S. P. Hong and M. M. Tripathi, On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci. 2 (2005), no. 2, 227–245
17. S. P. Hong and M. M. Tripathi, Ricci curvature of submanifolds of a Sasakian space form, Iranian J. Math. Sci. Inform. 1 (2006), no. 2, 31–51
18. J.-S. Kim, M. K. Dwivedi, and M. M. Tripathi, Ricci curvature of integral submanifolds of an S-space form, Bull. Korean Math. Soc. 44 (2007), no. 3, 395–406 https://doi.org/10.4134/BKMS.2007.44.3.395
19. M. Kobayashi, Semi-invariant submanifolds in an f-manifold with complemented frames, Tensor (N.S.) 49 (1990), no. 2, 154–177
20. M. Kobayashi and S. Tsuchiya, Invariant submanifolds of an f-manifold with complemented frames, Kodai Math. Sem. Rep. 24 (1972), 430–450 https://doi.org/10.2996/kmj/1138846636
21. I. Mihai, Ricci curvature of submanifolds in Sasakian space forms, J. Aust. Math. Soc. 72 (2002), no. 2, 247–256
22. I. Mihai, CR-submanifolds of a framed f-manifold, Stud. Cerc. Mat. 35 (1983), no. 2, 127–136
23. H. Nakagawa, On framed f-manifolds, Kodai Math. Sem. Rep. 18 (1966), 293–306 https://doi.org/10.2996/kmj/1138845274
24. B. Suceava, Some remarks on B.-Y. Chen's inequality involving classical invariants, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 45 (1999), no. 2, 405–412
25. M. M. Tripathi, Certain basic inequalities for submanifolds, Proceedings of the Tenth International Workshop on Differential Geometry, 99–145, Kyungpook Nat. Univ., Taegu, 2006
26. M. M. Tripathi and I. Mihai, Submanifolds of framed metric manifolds and S-manifolds, Note Mat. 20 (2000/01), no. 2, 135–164
27. M. M. Tripathi and K. D. Singh, Almost semi-invariant submanifolds of an $\epsilon$ -framed metric manifold, Demonstratio Math. 29 (1996), no. 2, 413–426
28. M. M. Tripathi and K. D. Singh, On submanifolds of S-manifolds, Ganita 47 (1996), no. 2, 51–54
29. J. Vanzura, Almost r-contact structures, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 97–115
30. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984
31. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, The curvature of submanifolds of an S-space form, Acta Math. Hungar. 62 (1993), no. 3-4, 373–383 https://doi.org/10.1007/BF01874657
32. B.-Y. Chen, On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms, Arch. Math. (Basel) 74 (2000), no. 2, 154–160 https://doi.org/10.1007/PL00000420
33. S. P. Hong, K. Matsumoto, and M. M. Tripathi, Certain basic inequalities for submanifolds of locally conformal Kaehler space forms, SUT J. Math. 41 (2005), no. 1, 75–94
34. T. Kashiwada, On a Riemannian manifold admitting a framed f - 3-structure, Natur. Sci. Rep. Ochanomizu Univ. 22 (1971), 91–99
35. M. M. Tripathi, On almost semi-invariant submanifolds, Ph. D. Thesis, Lucknow University, India, 1996
36. M. M. Tripathi, J.-S. Kim, and S. B. Kim, Mean curvature and shape operator of slant immersions in a Sasakian space form, Balkan J. Geom. Appl. 7 (2002), no. 1, 101–111

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