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SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM IN TERMS OF THE STRUCTURE JACOBI OPERATOR

  • Received : 2021.01.05
  • Accepted : 2021.05.06
  • Published : 2022.01.31
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Abstract

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form Mn+1(c), c ≠ 0. We denote by A and R𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(< 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R𝜉A = AR𝜉 and at the same time ∇𝜉R𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

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References

  1. A. Bejancu, CR submanifolds of a Kaehler manifold. I, Proc. Amer. Math. Soc. 69 (1978), no. 1, 135-142. https://doi.org/10.2307/2043207
  2. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141. https://doi.org/10.1515/crll.1989.395.132
  3. D. E. Blair, G. D. Ludden, and K. Yano, Semi-invariant immersions, Kodai Math. Sem. Rep. 27 (1976), no. 3, 313-319. http://projecteuclid.org/euclid.kmj/1138847256 https://doi.org/10.2996/kmj/1138847256
  4. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481-499. https://doi.org/10.2307/1998460
  5. T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015. https://doi.org/10.1007/978-1-4939-3246-7
  6. J. T. Cho and U.-H. Ki, Real hypersurfaces of a complex projective space in terms of the Jacobi operators, Acta Math. Hungar. 80 (1998), no. 1-2, 155-167. https://doi.org/10.1023/A:1006585128386
  7. J. T. Cho and U.-H. Ki, Real hypersurfaces in complex space forms with Reeb flow symmetric structure Jacobi operator, Canad. Math. Bull. 51 (2008), no. 3, 359-371. https://doi.org/10.4153/CMB-2008-036-7
  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. U.-H. Ki and H.-J. Kim, Semi-invariant submanifolds with lift-flat normal connection in a complex projective space, Kyungpook Math. J. 40 (2000), no. 1, 185-194.
  10. U.-H. Ki, H. Kurihara, S. Nagai, and R. Takagi, Characterizations of real hypersurfaces of type A in a complex space form in terms of the structure Jacobi operator, Toyama Math. J. 32 (2009), 5-23.
  11. U.-H. Ki, S. Nagai, and R. Takagi, The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms, Geom. Dedicata 149 (2010), 161-176. https://doi.org/10.1007/s10711-010-9474-y
  12. U.-H. Ki and H. Song, Jacobi operators on a semi-invariant submanifold of codimension 3 in a complex projective space, Nihonkai Math. J. 14 (2003), no. 1, 1-16.
  13. U.-H. Ki and H. Song, Semi-invariant submanifolds of codimension 3 in a complex space form with commuting structure Jacobi operator, to appear in Kyungpook Math. J.
  14. U.-H. Ki, H. Song, and R. Takagi, Submanifolds of codimension 3 admitting almost contact metric structure in a complex projective space, Nihonkai Math. J. 11 (2000), no. 1, 57-86.
  15. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149. https://doi.org/10.2307/2000565
  16. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986), no. 2, 245-261. https://doi.org/10.1007/BF00164402
  17. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, in Tight and taut submanifolds (Berkeley, CA, 1994), 233-305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997.
  18. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364. https://doi.org/10.2307/1998631
  19. M. Okumura, Codimension reduction problem for real submanifold of complex projective space, in Differential geometry and its applications (Eger, 1989), 573-585, Colloq. Math. Soc. Janos Bolyai, 56, North-Holland, Amsterdam, 1992.
  20. H. Song, Some differential-geometric properties of R-spaces, Tsukuba J. Math. 25 (2001), no. 2, 279-298. https://doi.org/10.21099/tkbjm/1496164288
  21. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka Math. J. 10 (1973), 495-506. http://projecteuclid.org/euclid.ojm/1200694557
  22. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures. I, II, J. Math. Soc. Japan 15, 27 (1975), 43-53, 507-516. https://doi.org/10.2969/jmsj/02710043
  23. Y. Tashiro, On the relationship between almost complex spaces and almost contact spaces centering around quasi-invariant subspaces of almost complex spaces, Sugaku 16 (1964), 54-64.
  24. K. Yano and U. H. Ki, On (f, g, u, v, w, λ, µ, ν)-structures satisfying λ2 + µ2 + ν2 = 1, Kodai Math. Sem. Rep. 29 (1977/78), no. 3, 285-307. http://projecteuclid.org/ euclid.kmj/1138833653 https://doi.org/10.2996/kmj/1138833653
  25. K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Progress in Mathematics, 30, Birkhauser, Boston, MA, 1983.