Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric
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Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric Hyunjin Lee1 · Young Jin Suh2 Received: 11 August 2020 / Accepted: 5 November 2020 / © Springer Nature B.V. 2020
Abstract In this paper, first, we investigate the commuting property between the normal Jacobi operator R¯ N and the structure Jacobi operator Rξ for Hopf real hypersurfaces in the complex quadric Qm = SOm+2 /SOm SO2 for m 3, which is defined by R¯ N Rξ = Rξ R¯ N . Moreover, a new characterization of Hopf real hypersurfaces with A-principal singular normal vector field in the complex quadric Qm is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric Qm with commuting Jacobi operators. Keywords Commuting Jacobi operator · A-isotropic · A-principal · K¨ahler structure · Complex conjugation · Complex quadric Mathematics Subject Classification (2010) Primary 53C40 · Secondary 53C55
1 Introduction In the class of Hermitian symmetric spaces of rank 2, usually we can give the examples of Riemannian symmetric spaces G2 (Cm+2 ) = SUm+2 /S(U2 Um ) and G∗2 (Cm+2 ) = SU2,m /S(U2 Um ), which are said to be complex two-plane Grassmannians and
The first author was supported by grant Proj. No. NRF-2019-R1I1A1A01050300 and the second author by NRF-2018-R1D1A1B05040381 from National Research Foundation of Korea. Hyunjin Lee
[email protected] Young Jin Suh [email protected] 1
The Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu 41566, Korea
2
Department of Mathematics & RIRCM, Kyungpook National University, Daegu 41566, Korea
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complex hyperbolic two-plane Grassmannians, respectively (see [4, 11, 24, 25] and [27]). These are viewed as Hermitian symmetric spaces and quaternionic K¨ahler symmetric spaces equipped with K¨ahler structure J and quaternionic K¨ahler structure J. There are exactly two types of singular tangent vectors X of complex 2-plane Grassmannians G2 (Cm+2 ) and complex hyperbolic 2-plane Grassmannians G∗2 (Cm+2 ) which are characterized by the geometric properties J X ∈ JX and J X ⊥ JX respectively. As another kind of Hermitian symmetric space with rank 2 of compact type different from the above ones, we can give the example of complex quadric Qm = SOm+2 /SOm SO2 , which is a complex hypersurface in complex projective space CP m+1 (see [20–23] and [26]). The complex quadric also can be regarded as a kind of real Grassmann manifold of compact type with rank 2 (see [7] and [12]). Accordingly, the complex quadric admits both a complex conjugation structure A and a K¨ahler structure J , which anti-commutes with each other, that is, AJ = −J A. Then for m 3 the triple (Qm , J, g) is a Hermitian symmetric space of compact type with rank 2 and its maximal sectional curvature is equal to 4 (see [10] and [20]). In addition to the K¨ahler structure J there is another distinguished geometric structure on Qm , namely a parallel rank two vector bundle A which
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