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GEOMETRIC INEQUALITIES FOR SUBMANIFOLDS IN SASAKIAN SPACE FORMS

  • Received : 2015.07.01
  • Published : 2016.07.31

Abstract

B. Y. Chen introduced a series of curvature invariants, known as Chen invariants, and proved sharp estimates for these intrinsic invariants in terms of the main extrinsic invariant, the squared mean curvature, for submanifolds in Riemannian space forms. Special classes of submanifolds in Sasakian manifolds play an important role in contact geometry. F. Defever, I. Mihai and L. Verstraelen [8] established Chen first inequality for C-totally real submanifolds in Sasakian space forms. Also, the differential geometry of slant submanifolds has shown an increasing development since B. Y. Chen defined slant submanifolds in complex manifolds as a generalization of both holomorphic and totally real submanifolds. The slant submanifolds of an almost contact metric manifolds were defined and studied by A. Lotta, J. L. Cabrerizo et al. A Chen first inequality for slant submanifolds in Sasakian space forms was established by A. Carriazo [4]. In this article, we improve this Chen first inequality for special contact slant submanifolds in Sasakian space forms.

Keywords

Acknowledgement

Supported by : European Social Found

References

  1. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, 2002.
  2. J. Bolton, F. Dillen, J. Fastenakels, and L. Vrancken, A best possible inequality for curvature like tensor fields, Math. Inequal. Appl. 12 (2009), no. 3, 663-681.
  3. J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), no. 1, 125-138. https://doi.org/10.1017/S0017089500010156
  4. A. Carriazo, A contact version of B.-Y. Chen's inequality and its applications to slant immersions, Kyungpook Math. J. 39 (1999), no. 2, 465-476.
  5. B. Y. Chen, Geometry of Slant Submanifolds, K. U. Leuven, 1990.
  6. B. Y. Chen, Pseudo-Riemannian Geometry, $\delta$-Invariants and Applications, World Scientific, 2011.
  7. D. Cioroboiu and A. Oiaga, B. Y. Chen inequalities for slant submanifolds in Sasakian space forms, Rend. Circ. Mat. Palermo (2) 52 (2003), no. 3, 367-381. https://doi.org/10.1007/BF02872761
  8. F. Defever, I. Mihai, and L. Verstraelen, B.-Y. Chen's inequality for C-totally real sub- manifolds of Sasakian space forms, Boll. Un. Mat. Ital. B (7) 11 (1997), no. 11, 365-374.
  9. A. Mihai, B. Y. Chen inequalities for slant submanifolds in generalized complex space forms, Rad. Mat. 12 (2004), no. 2, 215-231.
  10. A. Mihai, Geometric inequalities for purely real submanifolds in complex space forms, Results Math. 55 (2009), no. 3-4, 457-468. https://doi.org/10.1007/s00025-009-0429-2
  11. I. Mihai and V. Ghisoiu, Minimality of certain contact slant submanifolds in Sasakian space forms, Int. J. Pure Appl. Math. Sci. 1 (2004), 95-99.
  12. K. Yano and M. Kon, Structures on Manifolds, World Scientific, 1984.