본문 바로가기
KERD
논문KCI 등재

SEMI-INVARINAT SUBMANIFOLDS OF CODIMENSION 3 SATISFYING ∇ϕ∇ξξRξ = 0 IN A COMPLEX SPACE FORM

기우항
영남수학회 (Youngnam Mathematical Society)|2021.01|East Asian Mathematical Journal|제37권 제1호|pp. 41-77

2

조회

0

다운로드

1

인용

0

참고문헌

키워드

distinguished normalcomplex space formstructure Jacobi operatorSemi-invariant submanifoldalmost contact metric structureHopf hypersurface
주제분류수학

초록

Let $M$ be a semi-invariant submanifold of codimension 3with almost contact metric structure $(\phi, \xi, \eta, g)$ in a complex space form$M_{n+1} (c)$, $c\neq0$. We denote by $R_\xi =R(\cdot,\xi)\xi$ and $A^{(i)}$ beJacobi operator with respect to the structure vector field $\xi$ and be the secondfundamental form in the direction of the unit normal $C^{(i)}$, respectively. Suppose that the third fundamental form $t$ satisfies $dt(X,Y)=2\theta g(\phi X,Y)$ forcertain scalar $\theta$ ($\neq2c$)and any vector fields $X$ and $Y$ and at the same time $R_\xi$is $\phi\nabla_\xi \xi$-parallel, then $M$ is a Hopf hypersurface in $M_n (c)$ providedthat it satisfies $R_\xi A^{(1)} =A^{(1)} R_\xi$, $R_\xi A^{(2)} =A^{(2)} R_\xi$ and$\bar{r}-2(n-1)c\leq0$, where $\bar{r}$ denotes the scalar curvature of $M$.

Abstract

Let $M$ be a semi-invariant submanifold of codimension 3with almost contact metric structure $(\phi, \xi, \eta, g)$ in a complex space form$M_{n+1} (c)$, $c\neq0$. We denote by $R_\xi =R(\cdot,\xi)\xi$ and $A^{(i)}$ beJacobi operator with respect to the structure vector field $\xi$ and be the secondfundamental form in the direction of the unit normal $C^{(i)}$, respectively. Suppose that the third fundamental form $t$ satisfies $dt(X,Y)=2\theta g(\phi X,Y)$ forcertain scalar $\theta$ ($\neq2c$)and any vector fields $X$ and $Y$ and at the same time $R_\xi$is $\phi\nabla_\xi \xi$-parallel, then $M$ is a Hopf hypersurface in $M_n (c)$ providedthat it satisfies $R_\xi A^{(1)} =A^{(1)} R_\xi$, $R_\xi A^{(2)} =A^{(2)} R_\xi$ and$\bar{r}-2(n-1)c\leq0$, where $\bar{r}$ denotes the scalar curvature of $M$.

참고문헌 (33)

  1. A. Bejancu, "CR-submanifolds of a Kahler ¨ manifold I", Proc. Amer. Math. Soc, 69, pp.135~142 (1978)
  2. J. Berndt, "Real hypersurfaces with constant principal curvatures in a complex hyperbolic space", J. Reine Angew. Math, 395, pp.132~141 (1989)
  3. J. Berndt, "Cohomogeneity one actions on non compact symmetric spaces of rank one", Trans. Amer. Math. Soc, 359, pp.3425~3438 (2007)
  4. J. Berndt, "Two natural generalizations of locally symmetric spaces", Diff. Geom. Appl, 2, pp.57~82 (1992)
  5. D. E. Blair, "Semi-invariant immersion", Kodai Math. Sem. Rep, 27, pp.313~319 (1976)
  6. T. E. Cecil, "Focal sets and real hypersurfaces in complex projective space", Trans. Amer. Math. Soc, 269, pp.481~499 (1982)
  7. T. E. Cecil, "Geometry of Hypersurfaces" (2015), Springer
  8. J. T. Cho, "Real hypersurfaces of a complex projective space in terms of the Jacobi operators", Acta Math. Hungar, 80, pp.155~167 (1998)
  9. J. T. Cho, "Real hypersurfaces in complex space forms with Reeb flow symmetric Jacobi operator", Canadian Math. Bull, 51, pp.359~371 (2008)
  10. J. Erbacher, "Reduction of the codimension of an isometric immersion", J. Diff. Geom, 3, pp.333~340 (1971)
  11. 허종임, "Semi-invariant submanifolds of codimension 3 of a complex projective space in terms of the Jacobi operator", 대한수학회보, 42(1), pp.93~119 (2005), 대한수학회
  12. U-H. Ki, "Cyclic-parallel real hypersurfaces of a complex space form", Tsukuba J. Math, 12, pp.259~268 (1988)
  13. 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), pp.271~306 (2019)
  14. 기우항, "Structure Jacobi Operator of Semi-invarinat Submanifolds in Complex Space Forms", East Asian Mathematical Journal, 36(3), pp.389~415 (2020), 영남수학회
  15. 기우항, "The structure Jacobi Operator on real hypersurfaces in a nonflat complex space form", 대한수학회보, 42(2), pp.337~358 (2005), 대한수학회
  16. U-H. Ki, "Commuting structure Jacobi operator for real hypersurfaces in complex space forms", Advances in Pure Math., 3(2), pp.264~276 (2013)
  17. U-H. Ki, "Commuting structure Jacobi operator for real hypersurface in complex form II", Tsukurba J. Math, 42(2), pp.127~154 (2018)
  18. U-H. Ki, "Characterizations of real hypersurfaces of type A in a complex space form in terms of the structure Jacobi operator", Toyama Math. J, 32, pp.5~23 (2009)
  19. U-H. Ki, "Semi-invariant submanifolds of codimension 3 in a complex hyperbolic", Honam Math. J, 23, pp.91~111 (2001)
  20. U-H. Ki, "Jacobi operators on a semi-invariant submanifold of codemension 3 in a complex projective space", Nihonkai Math J, 14, pp.1~16 (2003)
  21. U-H. Ki, "Submanifolds of codimension 3 admitting almost contact metric structure in a complex projective space", Nihonkai Math J, 11, pp.57~86 (2000)
  22. M. Kimura, "Real hypersurfaces and complex submanifolds in complex projective space", Trans. Amer. Math. Soc, 296, pp.137~149 (1986)
  23. S. Montiel, "On some real hypersurfaces of a complex hyperbolic space", Geom. Dedicata, 20, pp.245~261 (1986)
  24. R. Niebergall, "Real hypersurfaces in complex space form, in Tight and Taut submanifolds", pp.233~305 (1998), Cambridge University Press
  25. M. Okumura, "On some real hypersurfaces of a complex projective space", Trans. Amer. Math. Soc, 212, pp.355~364 (1973)
  26. M. Okumura, "Normal curvature and real submanifold of the complex projective space", Geom. Dedicata, 7, pp.509~517 (1978)
  27. M. Ortega, "Non-existence of real hypersurface with parallel structure Jacobi operator in nonflat complex space forms", Rocky Mountain J. Math, 36, pp.1603~1613 (2006)
  28. H. Song, "Some differential-geometric properties of R-spaces", Tsukuba J. Math, 25, pp.279~298 (2001)
  29. R. Takagi, "On homogeneous real hypersurfaces in a complex projective space", Osaka J. Math, 19, pp.495~506 (1973)
  30. R. Takagi, "Real hypersurfaces in a complex projective space with constant principal curvatures I,II", J. Math. Soc. Japan, 27, pp.43~53 (1975)
  31. Y. Tashiro, "Relations between the theory of almost complex spaces and that of almost contact spaces (in Japanese)", Sugaku, 16, pp.34~61 (1964)
  32. K. Yano, "On (f, g, u, v, w, λ, µ, ν)-structure satisfying λ2 + µ2 + ν2 = 1", Kodai Math. Sem. Rep, 29, pp.285~307 (1978)
  33. K. Yano, "CR submanifolds of Kaehlerian and Sasakian manifolds" (1983), Birkhauser

관련 논문AI추천

SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM WITH ξ-PARALLEL STRUCTURE JACOBI OPERATOR기우항, 송현정영남수학회 · East Asian Mathematical Journal2024 · 제40권 제1호Semi-invariant submanifolds of codimension 3 in a complex space form in terms of the structure Jacobi operatorHiroyuki Kurihara, U-Hang Ki대한수학회 · 대한수학회논문집2022 · 제37권 제1호Structure Jacobi Operator of Semi-invarinat Submanifolds in Complex Space Forms기우항, 김수진영남수학회 · East Asian Mathematical Journal2020 · 제36권 제3호Submanifolds of Codimension 3 in a Complex Space Form with Commuting Structure Jacobi Operator기우항, 송현정경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2022 · 제62권 제1호Commuting Structure Jacobi Operator for Semi-Invariant Submanifolds of Codimension 3 in Complex Space Forms기우항, 송현정영남수학회 · East Asian Mathematical Journal2022 · 제38권 제5호Structure Jacobi Operators of Semi-invarinat Submanifolds in a Complex Space Form II기우항, 김수진영남수학회 · East Asian Mathematical Journal2022 · 제38권 제1호

이 간행물의 다른 논문 (11건)

On the Modification of Finite Field based S-Box김준교pp. 1-7On the Touchard polynomials and Multiplicative Plane Partitions김준교pp. 9-17The Bergman Metric and Related Bloch Spaces on the Exponentially Weighted Bergman Space변지수, 조홍래, 이한울pp. 19-32Insertion Property by Essential Ideals윤상조, 서연숙, 남상복pp. 33-40Simplified Approach to Valuation of Vulnerable Exchange Option under a Reduced-Form Model허정규, 전재기, 김건우pp. 79-85
원문 보기