Real Hypersurfaces in Complex Two-Plane Grassmannians

Berndt, Jürgen; Suh, Young Jin

doi:10.1007/s006050050018

Cited by 126 publications

(239 citation statements)

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“…Berndt and the present author [2] have proved the following theorem. In Theorem A the vector £ contained in the one-dimensional distribution [£] is said to be a Hopf vector when it becomes a principal vector for the shape operator A of M in G 2 ( C m + 2 ) .…”

Section: Introductionmentioning

confidence: 66%

“…The present author would like to express his sincere gratitude to the referee for his valuable comments and suggestions to develop the first version of the manuscript. In this section we summarise basic material about G2(C m+2 ), for details we refer to [2] and [3]. By G2(C m+2 ) we denote the set of all complex two-dimensional linear subspaces in C m + 2 .…”

Section: -mentioning

confidence: 99%

“…In the geometry of real hypersurfaces in complex space forms M m (c) or in quaternionic space forms there have been many characterisations of model hypersurfaces of type A\, A 2 , B, C, D and E in complex projective space C P m , of type A o , A X ,A 2 and B in complex hyperbolic space CH m or A\,A 2 ,B in quaternionic projective space QP m , which are completely classified by Cecil and Ryan [4], Kimura [5], Berndt [1], Martinez and Perez [7] respectively. Among them there were only a few characterisations of homogeneous real hypersurfaces of type B in complex projective space C P m .…”

Section: Introductionmentioning

confidence: 99%

“…It is the unique compact irreducible Riemannian manifold, being equipped with both a Kahler structure J and a quaternionic Kahler structure 3 not containing 380 Y.J. Suh [2] J. In other words, (^(C" 1 " 1 " 2 ) is the unique compact, irreducible, Kahler, quaternionic Kahler manifold which is not a hyper-Kahler manifold.…”

Section: Introductionmentioning

confidence: 99%

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Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator

Suh¹

2003

Bull. Austral. Math. Soc.

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In this paper we give a non-existence property of real hypersurfaces in complex twoplane Grassmannians G 2 (C m + 2 ) which have a shape operator A commuting with the structure tensors {0i, fc, satisfy A(/> + (j>A = k(j>, for some constant k, is a model characterisation of this kind for type B, which is a tube over a real projective space R P m in C P m (See Yano and Kon [8])-On the other hand, when we consider real hypersurfaces in quaternionic projective space Q P m , Pak [6] has considered a geometric condition that A u -$ V A = 0, v = 1,2,3 that is, the structure tensor 4> v and the shape operator A commute with each other. Moreover, it was known to be a characterisation of type A\, and A 2 in quaternionic projective space Q P m , which is a tube of radius r, 0 < r < n/2, over a totally geodesic QP k in QP m . Now let us denote by G2(C m+2 ) the set of all two-dimensional linear subspaces in C m + 2 . This Riemannian symmetric space G2(C m+2 ) has a remarkable geometrical structure. It is the unique compact irreducible Riemannian manifold, being equipped with both a Kahler structure J and a quaternionic Kahler structure 3 not containing

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Section: Introductionmentioning

confidence: 66%

Section: -mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator

Suh¹

2003

Bull. Austral. Math. Soc.

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“…So, in G 2 (C m+2 ) we have two natural geometrical conditions for real hypersurfaces that [ξ] = Span{ξ} or D ⊥ = Span{ξ 1 , ξ 2 , ξ 3 } is invariant under the shape operator. By using such kinds of geometric conditions Berndt and Suh [3] have proved the following: …”

Section: Introductionmentioning

confidence: 99%

Commuting Structure Jacobi Operator for Hopf Hypersurfaces in Complex Two-Plane Grassmannians

Jeong¹,

Suh²,

Yang³

2009

Bulletin of the Korean Mathematical Society

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Abstract. In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m+2 ) satisfying the condition that the structure Jacobi operator R ξ commutes with the 3-structure tensors φ i , i = 1, 2, 3. IntroductionIn the geometry of real hypersurfaces in complex space forms M n (c) Kimura [6] has proved that Hopf real hypersurfaces in a complex projective space P n (C) with constant principal curvatures are locally congruent to of type (A), a tube over a totally geodesic P k (C), of type (B), a tube over a complex quadric Q n−1 , cot 2 2r = n−2, of type (C), a tube over P 1 (C)×P (n−1)/2 (C), cot 2 2r = 1 n−2 and n is odd, of type (D), a tube over a complex two-plane Grassmannian G 2 (C 5 ), cot 2 2r = 3 5 and n = 9, of type (E), a tube over a Hermitian symmetric space SO(10)/U (5), cot 2 2r = 5 9 and n = 15. The notion of Hopf real hypersurfaces means that the structure vector ξ defined by ξ = −JN satisfies Aξ = αξ, where J denotes a Kaehler structure of P n (C), N and A a unit normal and the shape operator of M in P n (C).A Jacobi field along geodesics of a given Riemannian manifold (M, g) is an important role in the study of differential geometry. It satisfies an well-known differential equation which inspires Jacobi operators. The Jacobi operator is defined by (R X (Y ))(p) = (R(Y, X)X)(p), where R denotes the curvature tensor of M and X, Y denote tangent vector fields on M . Then we see that R X is a self-adjoint endomorphism on the tangent space of M and is related to the differential equation, so called Jacobi equation, which is given by ∇ γ (∇ γ Y ) +

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Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

Suh

Woo

2014

Mathematische Nachrichten

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Real Hypersurfaces in Complex Two-Plane Grassmannians

Cited by 126 publications

References 2 publications

Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator

Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator

Commuting Structure Jacobi Operator for Hopf Hypersurfaces in Complex Two-Plane Grassmannians

Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

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