Recent Results on Real Hypersurfaces in Complex Quadrics
In this survey article, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 2. Second, by using the isometric Reeb flow, we give a complete classification for hypersurfaces M in complex two-plane G
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Abstract In this survey article, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 2. Second, by using the isometric Reeb flow, we give a complete classification for hypersurfaces M in complex two-plane Grassmannians G 2 (Cm+2 ) = SU2+m /S(U2 Um ), complex hyperbolic two-plane Grassmannians G ∗2 (Cm+2 ) = SU2,m /S(U2 Um ), complex quadric o /S Om S O2 . As a third, we introQ m = SOm+2 /S Om S O2 and its dual Q m∗ = S Om,2 duce the classifications of contact hypersurfaces with constant mean curvature in the complex quadric Q m and its noncompact dual Q m∗ for m ≥ 3. Finally we want to mention some classifications of real hypersurfaces in the complex quadrics Q m with Ricci parallel, harmonic curvature, parallel normal Jacobi, pseudo-Einstein, pseudo-anti commuting Ricci tensor and Ricci soliton etc.
1 Introduction ¯ g) a Riemannian manifold and I ( M, ¯ g) the set of all isomeLet us denote by ( M, ¯ ¯ g) is a connected tries defined on M. Here, a homogeneous submanifold of ( M, ¯ ¯ g). If submanifold M of M which is an orbit of some closed subgroup G of I ( M, the codimension of M is one, then M is called a homogeneous hypersurface. When ¯ there exists some closed subgroup M becomes a homogeneous hypersurface of M, ¯ G of I ( M, g) having M as an orbit. Since the codimension of M is one, the regular orbits of the action of G on M¯ have codimension one, that is, the action of G on M¯ is of cohomogeneity one. This means that the classification of homogeneous hypersurfaces is equivalent to the classification of cohomogeneity one actions up to orbit equivalence. ¯ ¯ g) The orbit space M/G with quotient topology for a closed subgroup G of I ( M, with cohomogeneity one becomes a one dimensional Hausdorff space homeomorphic Y.J. Suh (B) 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_28
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to the real line R, the circle S 1 , the half-open interval [0, ∞), or the closed interval [0, 1]. This was proved by Mostert [28] for the case G is compact and in general by Bérard-Bergery. ¯ When M¯ is simply connected and compact, the quotient space M/G must be homeomorphic to [0, 1] and each singular orbit must have codimension greater than one. This means that each regular orbit is a tube around any of the two singular orbits, and each singular orbit is a focal set of any regular orbit. This fact will be applied in Sects. 2, 4 and 6 for complex projective space CP m , complex two-plane Grassmannians G 2 (Cm+2 ) and complex quadric Q m which are Hermitian symmetric spaces of compact type with rank 1 and rank 2 respectively. ¯ When M¯ is simply connected and non-compact, the quotient space M/G must be homeomorphic to R or [0, ∞). In the latter case the singular orbit must have codimension greater th
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