Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RICCI CURVATURE OF SUBMANIFOLDS OF AN S-SPACE FORM

  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

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

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, 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
  8. 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
  9. J. L. Cabrerizo, L. M. Fernandez, and M. Fernandez, On normal CR-submanifolds of S-manifolds, Colloq. Math. 64 (1993), no. 2, 203–214
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. B.-Y. Chen, Riemannian submanifolds, Handbook of differential geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000
  16. 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
  17. 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
  18. 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
  19. S. P. Hong and M. M. Tripathi, On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci. 2 (2005), no. 2, 227–245
  20. 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
  21. T. Kashiwada, On a Riemannian manifold admitting a framed f - 3-structure, Natur. Sci. Rep. Ochanomizu Univ. 22 (1971), 91–99
  22. 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
  23. M. Kobayashi, Semi-invariant submanifolds in an f-manifold with complemented frames, Tensor (N.S.) 49 (1990), no. 2, 154–177
  24. 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
  25. I. Mihai, CR-submanifolds of a framed f-manifold, Stud. Cerc. Mat. 35 (1983), no. 2, 127–136
  26. I. Mihai, Ricci curvature of submanifolds in Sasakian space forms, J. Aust. Math. Soc. 72 (2002), no. 2, 247–256
  27. H. Nakagawa, On framed f-manifolds, Kodai Math. Sem. Rep. 18 (1966), 293–306 https://doi.org/10.2996/kmj/1138845274
  28. 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
  29. M. M. Tripathi, On almost semi-invariant submanifolds, Ph. D. Thesis, Lucknow University, India, 1996
  30. M. M. Tripathi, Certain basic inequalities for submanifolds, Proceedings of the Tenth International Workshop on Differential Geometry, 99–145, Kyungpook Nat. Univ., Taegu, 2006
  31. 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
  32. M. M. Tripathi and I. Mihai, Submanifolds of framed metric manifolds and S-manifolds, Note Mat. 20 (2000/01), no. 2, 135–164
  33. 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
  34. M. M. Tripathi and K. D. Singh, On submanifolds of S-manifolds, Ganita 47 (1996), no. 2, 51–54
  35. J. Vanzura, Almost r-contact structures, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 97–115
  36. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984

Cited by

  1. Chen-Tripathi Inequality for Warped Product Submanifolds of S-space Forms vol.58, pp.1, 2012, https://doi.org/10.2478/v10157-011-0050-z