Reeb Recurrent Structure Jacobi Operator on Real Hypersurfaces in Complex Two-Plane Grassmannians
In (Jeong et al., Acta Math Hungar 122(1–2), 173–186, 2009) [7 ], Jeong, Pérez, and Suh verified that there does not exist any connected Hopf hypersurface in complex two-plane Grassmannians with parallel structure Jacobi operator. In this paper, we consid
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Abstract In (Jeong et al., Acta Math Hungar 122(1–2), 173–186, 2009) [7], Jeong, Pérez, and Suh verified that there does not exist any connected Hopf hypersurface in complex two-plane Grassmannians with parallel structure Jacobi operator. In this paper, we consider more general notions as Reeb recurrent or Q ⊥ -recurrent structure Jacobi operator. By using these general notions, we give some new characterizations of Hopf hypersurfaces in complex two-plane Grassmannians.
1 Introduction As examples of Hermitian symmetric spaces of rank 2 we can give Riemannian symmetric spaces SU2,m /S(U2 Um ) and S Om+2 /S Om S O2 , which are said to be complex hyperbolic two-plane Grassmannians and complex quadric, respectively. Recently, the second author have studied hypersurfaces of those spaces (see [19–22]). On the other hand, as another kind of Hermitian symmetric spaces with rank 2 of compact type, we have complex two-plane Grassmannians G 2 (Cm+2 ) which are the sets of all complex two-dimensional linear subspaces in Cm+2 . Riemannian symmetric space G 2 (Cm+2 ) has a remarkable geometric structure. It is the unique compact irreducible Riemannian manifold being equipped with both a Kaehler structure J and a quaternionic Kaehler structure J not containing J , for details we refer to [1, 2, 15–18]. In particular, when m = 1, G 2 (C3 ) is isometric to the two-dimensional complex projective space CP 2 with constant holomorphic sectional curvature eight. When m = 2, we note that the isomorphism Spin(6) SU(4) yields an isometry between H. Lee (B) Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu 41566, Republic of Korea e-mail: [email protected] Y.J. Suh Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 Y.J. Suh et al. (eds.), Hermitian–Grassmannian Submanifolds, Springer Proceedings in Mathematics & Statistics 203, DOI 10.1007/978-981-10-5556-0_7
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6 G 2 (C4 ) and the real Grassmann manifold G + 2 (R ) of oriented two-dimensional linear 6 subspaces in R . Hereafter, we will assume m ≥ 3. On a real hypersurface M in G 2 (Cm+2 ), the almost contact structure vector field ξ defined by ξ = −J N is said to be the Reeb vector field, where N denotes a local unit normal vector field to M in G 2 (Cm+2 ). And a real hypersurface such that A[ξ ] ⊂ [ξ ] is called Hopf hypersurface. The almost contact 3-structure vector fields ξν for the 3-dimensional distribution Q ⊥ of M in G 2 (Cm+2 ) are defined by ξν = −Jν N (ν = 1, 2, 3), where {Jν }ν=1,2,3 denotes a canonical local basis of a quaternionic Kaehler structure J, such that Tx M = Q ⊕ Q ⊥ , x ∈ M. In addition, a real hypersurface of G 2 (Cm+2 ) satisfying g(AQ, Q ⊥ ) = 0 (i.e. AQ ⊥ ⊂ Q ⊥ or AQ ⊂ Q, resp.) is said to be a Q ⊥ -invariant hypersurface. We can naturally consider two geometric conditions that the 1-dimensional distribution [ξ ] = Span{ξ } and the 3-dimensional distribution Q ⊥ = Span{ξ1
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