East Asian mathematical journal

  • 제38권1호
  • /
  • Pages.43-63
  • /
  • 2022
  • /
  • 1226-6973(pISSN)
  • /
  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

STRUCTURE JACOBI OPERATORS OF SEMI-INVARINAT SUBMANIFOLDS IN A COMPLEX SPACE FORM II

  • Ki, U-Hang (The National Academy of Sciences) ;
  • Kim, Soo Jin (Department of Mathematics Chosun University Gwangju)
  • 투고 : 2021.11.29
  • 심사 : 2021.12.31
  • 발행 : 2022.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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). We denote by Rξ the structure Jacobi operator with respect to the structure vector field ξ and by ${\bar{r}}$ the scalar curvature of M. Suppose that Rξ is φ∇ξξ-parallel and at the same time 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 Rξφ = φRξ, then M is a Hopf hypersurface of type (A) in Mn+1(c) provided that ${\bar{r}-2(n-1)c}$ ≤ 0.

키워드

참고문헌

  1. A. Bejancu, CR-submanifolds of a Kahler manifold I, Proc. Amer. Math. Soc. 69(1978), 135-142. https://doi.org/10.1090/S0002-9939-1978-0467630-0
  2. J. Berndt, Real hypersurfaces with constant principal curvatures in a complex hyperbolic space, J. Reine Angew. Math. 395(1989), 132-141.
  3. J. Berndt and H. Tamaru, Cohomogeneity one actions on non compact symmetric spaces of rank one, Trans. Amer. Math. Soc. 359 (2007), 3425-3438. https://doi.org/10.1090/S0002-9947-07-04305-X
  4. D. E. Blair, G. D. Ludden and K. Yano, Semi-invariant immersions, Kodai Math. Sem. Rep. 27(1976), 313-319. https://doi.org/10.2996/kmj/1138847256
  5. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269(1982), 481-499. https://doi.org/10.1090/S0002-9947-1982-0637703-3
  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), 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 Jacobi operator, Canadian Math. Bull. 51(2008), 359-371. https://doi.org/10.4153/CMB-2008-036-7
  8. J. Erbacher, Reduction of the codimension of an isometric immersion, J. Diff. Geom. 3(1971), 333-340.
  9. J. I. Her, U-H. Ki and S.-B. Lee, Semi-invariant submanifolds of codimension 3 of a complex projective space in terms of the Jacobi operator, Bull. Korean Math. Soc. 42(2005), 93-119. https://doi.org/10.4134/BKMS.2005.42.1.093
  10. S. Kawamoto, Codimension reduction for real submanifolds of a complex hyperbolic space. Rev. math. univ. comput. madrid. 7(1994), 119-128.
  11. U-H. Ki, Cyclic-parallel real hypersurfaces of a complex space form, Tsukuba J. Math. 12(1988), 259-268.
  12. U-H. Ki, Commuting Jacobi operators for a semi-invariant submanifold of codimension 3 in complex space forms, J. Nat. Acad. Sci. ROK, Sci. Ser. 58-2(2019), 271-306.
  13. Ki, U-H, The Ricci tensor and structure Jacobi operator of submanifolds of codimension 3 in complex space forms. J. Nat. Acad. Sci. ROK, Sci. Ser. 59-2(2020), 367-394.
  14. Ki, U-H. and Kim, S. J, Jacobi operators along the structure flow on semi-invariant subminifolds of codimension 3 in a complex space form. J. Nat. Acad. Sci. ROK. Sci. Ser. 59-1(2020), 19-44.
  15. U-H. Ki and S. J. Kim, Structure Jacobi operators of semi-invariant submanifolds in a complex space form, East Asian Math. J. 36-3(2020), 389-415. https://doi.org/10.7858/EAMJ.2020.027
  16. U-H. Ki, S. J. Kim and H. Kurihara, Jacobi operators with respect to the Reeb vector fields on real hypersurfaces in a nonflat complex space form, Kyungpook Math J. 56(2016), 541-575. https://doi.org/10.5666/KMJ.2016.56.2.541
  17. U-H. Ki and H. Kurihara, Semi-invariant submanifolds of codimension 3 in a complex space form in terms of the structure Jacobi operators, to appear in Comm. Korean Math. Soc.
  18. U-H. Ki, S. Nagai and R. Takagi, The structure vector filed 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
  19. U-H. Ki and H. Song, Jacobi operators on a semi-invariant submanifolds of codimension 3 in a complex projective space, Nihonkai Math J. 14(2003), 1-16.
  20. U-H. Ki and H. Song, Submanifold of codimension 3 with almost contact metric structure in a complex space form, J. Nat. Acad. Sci. ROK. Sci. Ser. 58-1(2019), 37-63.
  21. U-H. Ki and H. Song, Semi-invariant submanifolds of condimension 3 in a complex space form with commuting structure Jacobi operator, to appear in Kyungpook. Math. J.
  22. 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), 57-86.
  23. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296(1986), 137-149. https://doi.org/10.1090/S0002-9947-1986-0837803-2
  24. S. Montiel and A.Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20(1986), 245-261. https://doi.org/10.1007/BF00164402
  25. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space form, in Tight and Taut submanifolds, Cambridge University Press : (1998(T. E. Cecil and S.-S. Chern eds.)), 233-305.
  26. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212(1973), 355-364. https://doi.org/10.2307/1998631
  27. M. Okumura, Normal curvature and real submanifold of the complex projective space. Geom. Dedicata 7(1978), 509-517. https://doi.org/10.1007/BF00152072
  28. M. Okumura, Codimension reduction problem for real submanifolds of complex projective space. Colloq. Math Janos Bolyai. 56(1989), 574-585.
  29. M. Ortega, J. D. Perez and F. G. Santos, Non-existence of real hypersurface with parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. Math. 36(2006), 1603-1613. https://doi.org/10.1216/rmjm/1181069385
  30. H. Song, Some differential-geometric properties of R-spaces, Tsukuba J. Math. 25(2001), 279-298.
  31. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19(1973), 495-506.
  32. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I,II, J. Math. Soc. Japan 27(1975), 43-53, 507-516. https://doi.org/10.2969/jmsj/02710043
  33. Y. Tashiro, Relations between the theory of almost complex spaces and that of almost contact spaces (in Japanese), Sugaku 16(1964), 34-61.
  34. K. Yano, and U-H. Ki, On (f, g, u, v, w, λ, µ, ν)-structure satisfying λ2 + µ2 + ν2 = 1, Kodai Math. Sem. Rep. 29(1978), 285-307. https://doi.org/10.2996/kmj/1138833653
  35. K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Birkhauser (1983).