Bulletin of the Korean Mathematical Society (대한수학회보)

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RICCI CURVATURE OF INTEGRAL SUBMANIFOLDS OF AN S-SPACE FORM

  • Published : 2007.08.31
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Abstract

Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for an integral submanifold of an S-space form. By polarization, we get a basic inequality for Ricci tensor also. Equality cases are also discussed. By giving a very simple proof we show that if an integral submanifold of maximum dimension of an S-space form satisfies the equality case, then it must be minimal. These results are applied to get corresponding results for C-totally real submanifolds of a Sasakian space form and for totally real submanifolds of a complex space form.

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References

  1. D. E. Blair, Geometry of manifolds with structural group $U(n){\times}O(s)$, J. Diff. Geometry 4 (1970), 155-167 https://doi.org/10.4310/jdg/1214429380
  2. D. E. Blair, On a generalization of the Hopf fibration, An. Sti. Univ. 'Al. I. Cuza' Iasi Sect. I a Mat. (N.S.) 17 (1971), 171-177
  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2002
  4. 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/10.1090/S0002-9947-1973-0319099-4
  5. 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
  6. 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
  7. 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
  8. B.-Y. Chen, On Ricci curvature of isotropic and Langrangian submanifolds in complex space forms, Arch. Math. (Basel) 74 (2000), no. 2, 154-160 https://doi.org/10.1007/PL00000420
  9. S. P. Hong and M. M. Tripathi, On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci. 2 (2005), no. 2, 227-245
  10. X. Liu, On Ricci curvature of C-totally real submanifolds in Sasakian space forms, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 17 (2001), no. 3, 171-177
  11. K. Matsumoto, I. Mihai, Ricci tensor of C-totally real submanifolds in Sasakian space forms, Nihonkai Math. J. 13 (2002), no. 2, 191-198
  12. I. Mihai, Ricci curvature of submanifolds in Sasakian space forms, J. Aust. Math. Soc. 72 (2002), no. 2, 247-256 https://doi.org/10.1017/S1446788700003888
  13. H. Nakagawa, On framed f-manifolds, Kodai Math. Sem. Rep. 18 (1966) 293-306 https://doi.org/10.2996/kmj/1138845274
  14. D. Van Lindt, P. Verheyen, and L. Verstraelen, Minimal submanifolds in Sasakian space forms, J. Geom. 27 (1986), no. 2, 180-187 https://doi.org/10.1007/BF01224555
  15. J. Van.zura, Almost r-contact structures, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 97-115
  16. S. Yamaguchi, M. Kon, and T. Ikawa, C-totally real submanifolds, J. Differential Geometry 11 (1976), no. 1, 59-64 https://doi.org/10.4310/jdg/1214433297
  17. K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984

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