Imsoon Jeong

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH 𝔇⊥-PARALLEL STRUCTURE JACOBI OPERATOR

In this paper we give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with 𝔇⊥-parallel structure Jacobi operator, where 𝔇⊥ = Span {ξ1, ξ2, ξ3}.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH LIE ξ-PARALLEL NORMAL JACOBI OPERATOR

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian G2( Cm+2) with Lie parallel normal Jacobi operator � RN and totally geodesic D and D? components of the Reeb ow.

Real Hypersurfaces in Complex Two-plane Grassmannians with F-parallel Normal Jacobi Operator

In this paper we give a non-existence theorem for Hopf hypersurfaces M in complex two-plane Grassmannians whose normal Jacobi operator is parallel on the distribution F defined by , where [] = Span{}, = Span {, , } and , .

RECURRENT JACOBI OPERATOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

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.

Real hypersurfaces of type (A) in complex two-plane Grassmannians related to the commuting shape operator

Abstract. We give some characterizations of real hypersurfaces of type (A) in complex two plane Grassmannians , that is, a tube over a totally geodesic in with the commuting condition for the shape operator A, the structure tensors and , together with additional geometric conditions.

Real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type

We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G2(ℂm+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇⊥-parallel for a hypersurface M in G2(ℂm+2), and give a characterization for a tube around a totally geodesic ℍ Pn in G2(ℂm+2) where m = 2n.

HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH LIE PARALLEL NORMAL JACOBI OPERATOR

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian G2(C) with Lie parallel normal Jacobi operator R̄N and totally geodesic D and D ⊥ components of the Reeb flow. 0. Introduction The Jacobi fields along geodesics of a given Riemannian manifold (M̄, ḡ) play an important role in the study of differential geometry. It satisfies a very well-known differential equation. This classical differential equation naturally inspires the so-called Jacobi operators. That is, if R̄ is the curvature operator of M̄ and X is any vector field tangent to M̄ , the Jacobi operator with respect to X at x ∈ M̄ , R̄X ∈ End(TxM̄), is defined as R̄X(Y )(x) = (R̄(Y,X)X)(x) for all Y ∈ TxM̄ , being a self-adjoint endomorphism of the tangent bundle TM̄ of M̄ . Clearly, each vector field X tangent to M̄ provides a Jacobi operator with respect to X (See [7] and [9]). If the structure vector field ξ = −JN of a real hypersurface M in complex projective space Pn(C) is invariant under the shape operator, ξ is said to be Hopf, where J denotes a Kähler structure of Pn(C), and N is a unit normal vector field of M in Pn(C). In the quaternionic projective space HP Pérez and Suh [10] classified the real hypersurfaces in HP with D⊥-parallel curvature tensor ∇ξνR = 0 for ν = 1, 2, 3, where R denotes the curvature tensor of M in HP and D⊥ is a distribution defined by D⊥ = Span {ξ1, ξ2, ξ3}. In this case they are congruent to a tube of radius π4 over a totally geodesic quaternionic submanifold HP k in HP, 2 ≤ k ≤ m− 2. Received September 22, 2009; Revised July 2, 2010. 2010 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

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

In this paper we give a characterization of a real hypersurface of Type (A) in complex two-plane Grassmannians G2(C ), which means a tube over a totally geodesic G2(C ) in G2(C ), by the Reeb parallel structure Jacobi operator∇ξRξ = 0.

ANTI-COMMUTING REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

In this paper we give a nonexistence theorem for real hypersurfaces in complex two-plane Grassmannians G2.C mC2 / with anti-commuting shape operator.