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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.

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