Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

doi:10.1016/J.DIFGEO.2007.11.015

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Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

Journal Article•DOI•

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

Juan de Dios Pérez¹, Florentino G. Santos¹•Institutions (1)

University of Granada¹

01 Apr 2008-Differential Geometry and Its Applications (North-Holland)-Vol. 26, Iss: 2, pp 218-223

TL;DR: In this paper, the authors classify real hypersurfaces of complex projective space C P m, m ⩾ 3, with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurface with recurrent structure JacobI operator.

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Abstract: We classify real hypersurfaces of complex projective space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.

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Journal Article•DOI•

Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie D -parallel

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Juan de Dios Pérez, Young Jin Suh

01 Jun 2013-Canadian Mathematical Bulletin

TL;DR: In this article, real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel are classified as hypersurface in the projective projective space, where the Jacobi operators are Lie parallel.

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Abstract: We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in

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23 citations

Cites background from "Real hypersurfaces in complex proje..."

...Also in [10], [11], [12], [13] we have studied distinct conditions on the structure Jacobi operator (Lie parallelism, Lie ξparallelism, D-parallelism, and so on)....
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Journal Article•DOI•

Real hypersurfaces in the complex quadric with parallel normal Jacobi operator

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Young Jin Suh¹•Institutions (1)

Kyungpook National University¹

01 Feb 2017-Mathematische Nachrichten

TL;DR: In this article, the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric was introduced and a complete classification of hypersurface with parallel normal J operator was given.

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Abstract: First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.

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13 citations

Cites background from "Real hypersurfaces in complex proje..."

...[9] J. D. Pérez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ -parallel, Differential Geom....
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...On the other hand, in [8,9] we have introduced the notion of structure Jacobi operator Rξ , which is a symmetric operator for real hypersurfaces in the complex projective space CPm , and have used it to study some principal curvatures for a tube over a totally geodesic submanifold (see Berger [1], Klein [3] and Reckziegel [10])....
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...Moreover, Pak, Suh and Woo [6] have focused on the study of commuting Jacobi operators, and Pérez and Santos [8], Pérez, Santos and Suh [9] respectively have investigated recurrent structure Jacobi operator ∇X Rξ = β(X)Rξ or Lie ξ -parallel structure Jacobi operator in CPm , that is, Lξ Rξ = 0 for any vector field X on a hypersurface M in CPm ....
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...Moreover, Pak, Suh and Woo [6] have focused on the study of commuting Jacobi operators, and Pérez and Santos [8], Pérez, Santos and Suh [9] respectively have investigated recurrent structure Jacobi operator ∇X Rξ = β(X)Rξ or Lie ξ -parallel structure Jacobi operator in CPm , that is, Lξ Rξ = 0 for any vector field X on a hypersurface M in CPm ....
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...[8] J. D. Pérez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Differential Geom....
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Journal Article•DOI•

Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator Is of Codazzi Type

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Juan de Dios Pérez¹, Florentino G. Santos¹, Young Jin Suh²•Institutions (2)

University of Granada¹, Kyungpook National University²

01 Sep 2007-Canadian Mathematical Bulletin

TL;DR: In this paper, it was shown that there is no real hypersurface in complex projective space whose structure Jacobi operator is of Codazzi type, and hence, no hypersurfaces in projective spaces.

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Abstract: We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type

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10 citations

Journal Article•DOI•

Recurrent jacobi operator of real hypersurfaces in complex two-plane grassmannians

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Imsoon Jeong, Juan de Dios Pérez, Young Jin Suh

31 Mar 2013-Bulletin of The Korean Mathematical Society

TL;DR: In this paper, the authors give a non-existence theorem for Hopf hy-persurfaces in the complex two-plane Grassmannian G2(C m+2 ) with re-current normal Jacobi operator ¯ RN.

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Abstract: In this paper we give a non-existence theorem for Hopf hy- persurfaces in the complex two-plane Grassmannian G2(C m+2 ) with re- current normal Jacobi operator ¯ RN.

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10 citations

Cites background from "Real hypersurfaces in complex proje..."

...For a real hypersurface in complex projective space CP, Pérez and Santos [15] introduced a new notion of D-recurrent, which is weaker than the structure Jacobi operator being recurrent....
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...[15] J. D. Pérez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Differential Geom....
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...For a real hypersurface in complex projective space CPm, Pérez and Santos [15] introduced a new notion of D-recurrent, which is weaker than the structure Jacobi operator being recurrent....
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Book Chapter•DOI•

Real Hypersurfaces in Complex Two-Plane Grassmannians with Recurrent Structure Jacobi Operator

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Imsoon Jeong¹, Young Jin Suh¹, Changhwa Woo¹•Institutions (1)

Kyungpook National University¹

01 Jan 2014

TL;DR: In this paper, a new notion of recurrent structure Jacobi operator was introduced for tangent vector fields X and Y on a real hypersurface M in a complex two-plane Grassmannian.

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Abstract: In this paper, we introduce a new notion of recurrent structure Jacobi operator, that is, \(( abla _{X}R_{\xi })Y =\omega (X)R_{\xi }Y\) for any tangent vector fields X and Y on a real hypersurface M in a complex two-plane Grassmannian, where R ξ denotes the structure Jacobi operator and ω a certain 1-form on M. Next, we prove that there does not exist any Hopf hypersurface M in a complex two-plane Grassmannian with recurrent structure Jacobi operator.

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8 citations

Cites background from "Real hypersurfaces in complex proje..."

...From such a view point, Pérez and Santos [8] have defined the notion of recurrent structure Jacobi operator in CP defined in such a way that ....
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...Using such a notion, they [8] proved that there does not exist any hypersurface in CP with recurrent structure Jacobi operator....
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References

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Book•

Foundations of Differential Geometry

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Shoshichi Kobayashi, Katsumi Nomizu

01 Jan 1963

7,658 citations

Additional excerpts

...This notion generalizes the fact of T being parallel, see [9]....
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Journal Article•DOI•

On homogeneous real hypersurfaces in a complex projective space

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Ryoichi Takagi

01 Dec 1973-Osaka Journal of Mathematics

TL;DR: In this paper, the authors considered the problem of determining homogeneous real hypersurfaces in a complex projective space Pn(C) of complex dimension n(^>2) which are orbits under analytic subgroups of the projective unitary group PU(n-\\-\\)>.

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Abstract: The purpose of this paper is to determine those homogeneous real hypersurfaces in a complex projective space Pn(C) of complex dimension n(^>2) which are orbits under analytic subgroups of the projective unitary group PU(n-\\-\\)> and to give some characterizations of those hypersurfaces. In § 1 from each effective Hermitian orthogonal symmetric Lie algebra of rank two we construct an example of homogeneous real hypersurface in Pn(C)y which we shall call a model space in Pn(C). In §2 we show that the class of all homogeneous real hypersurfaces in Pn{C) that are orbits under analytic subgroups of PU(n-\\-l) is exhausted by all model spaces. In §§3 and 4 we give some conditions for a real hypersurface in Pn(C) to be an orbit under an analytic subgroup of PU(n-\\-l) and in the course of proof we obtain a rigidity theorem in Pn(C) analogous to one for hypersurfaces in a real space form. The author would like to express his hearty thanks to Professor T. Takahashi for valuable discussions with him and his constant encouragement, and to Professor M. Takeuchi who made an original complicated proof of Lemma 2.3 short and clear.

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316 citations

Journal Article•DOI•

Real hypersurfaces in a complex projective space with constant principal curvatures II

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Ryoichi Takagi¹•Institutions (1)

University of Education, Winneba¹

01 Jan 1975-Journal of The Mathematical Society of Japan

229 citations

Journal Article•DOI•

On real hypersurfaces of a complex projective space

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Yoshiaki Maeda¹•Institutions (1)

Tokyo Metropolitan University¹

01 Jul 1976-Journal of The Mathematical Society of Japan

206 citations

"Real hypersurfaces in complex proje..." refers background in this paper

...In the sequel we need the following results, see [11]:...
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Journal Article•DOI•

A curvature characterization of certain locally rank-one symmetric spaces

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Quo-Shin Chi

01 Jan 1988-Journal of Differential Geometry

176 citations

"Real hypersurfaces in complex proje..." refers background in this paper

...For instance, in [1], it is pointed out that (locally) symmetric spaces of rank 1 (among them complex space forms) satisfy that all the eigenvalues of R̃X have constant multiplicities and are independent of the point and the tangent vector X....
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