Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CODIMENSION REDUCTION FOR SUBMANIFOLDS OF UNIT (4m+3)-SPHERE AND ITS APPLICATIONS

  • Received : 2013.03.07
  • Published : 2014.09.30
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Abstract

In this paper we establish codimension reduction theorem for submanifolds of a (4m+3)-dimensional unit sphere $S^{4m+3}$ with Sasakian 3-structure and apply it to submanifolds of a quaternionic projective space.

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References

  1. C. B. Allendoerfer, Rigidity for spaces of class greater than one, Amer. J. Math. Soc. 61 (1939), 633-644. https://doi.org/10.2307/2371317
  2. T. E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math J. 55 (1974), 5-31. https://doi.org/10.1017/S0027763000016202
  3. B. Y. Chen, Geometry of Submanifolds, Marcel Dekker Inc., New York, 1973.
  4. J. Erbacher, Reduction of the codimension of an isometric immersion, J. Differential Geometry 5 (1971), 333-340. https://doi.org/10.4310/jdg/1214429997
  5. S. Ishihara, Quaternion Kaehlerian manifolds, J. Differential Geometry 9 (1974), 483-500. https://doi.org/10.4310/jdg/1214432544
  6. S. Ishihara and M. Konish, Differential Geometry of Fibred Spaces, Study Group of Differential Geometry, Tokyo, 1973.
  7. T. Kashiwada, A note on a Riemannian space with Sasakian 3-structure, Natur. Sci. Rep. Ochanomizu Univ. 22 (1971), 1-2.
  8. S. Kawamoto, Codimension reduction for real submanifolds of a complex hyperbolic space, Rev. Mat. Univ. Complut. Madrid 7 (1994), no. 1, 119-128.
  9. H. S. Kim and J. S. Pak, Codimension reduction for real submanifolds of quaternionic hyperbolic space, Acta Math. Hungar. 121 (2008), no. 1-2, 21-33. https://doi.org/10.1007/s10474-008-7167-x
  10. Y. Y. Kuo, On almost contact 3-structure, Tohoku Math. J. 22 (1970), 325-332. https://doi.org/10.2748/tmj/1178242759
  11. J.-H. Kwon and J. S. Pak Codimension reduction for real submanifolds of quaternionic projective space, J. Korean Math. Soc. 36 (1999), no. 1, 109-123.
  12. H. B. Lawson, Jr., Rigidity theorems in rank-1 symmetric spaces, J. Differential Geom-etry 4 (1970), 349-357. https://doi.org/10.4310/jdg/1214429508
  13. M. Okumura, Reducing the codimension of a submanifold of a complex projective space, Geom. Dedicata 13 (1982), no. 3, 277-289.
  14. M. Okumura, Codimension reduction problem for real submanifold of complex projective space, Differential geometry and its applications (Eger, 1989), 573-585, Colloq. Math. Soc. Janos Bolyai, 56, North-Holland, Amsterdam, 1992.
  15. J. S. Pak, Real hypersurfaces in quaternionic Kaehlerian manifolds with constant Q-sectional curvature, Kodai Math. Sem. Rep. 29 (1977), no. 1-2, 22-61. https://doi.org/10.2996/kmj/1138833571
  16. Y. Shibuya, Real submanifolds in a quaternionic projective space, Kodai Math. J. 1 (1978), no. 3, 421-439. https://doi.org/10.2996/kmj/1138035651
  17. S. Tachibana and W. N. Yu, On a Riemannian space admitting more than one Sasakian structures, Tohoku Math. J. 22 (1970), 536-540. https://doi.org/10.2748/tmj/1178242720