Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians
type: Item
instance of: scholarly article
title: Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians
zbMATH Open document ID: 1015.53034
author: Young Jin Suh
author: Jürgen Berndt
published in: Monatshefte für Mathematik
publication date: 2002-12-01T00:00:00Z
review text: Let \((\overline M,g,J)\) be an almost Hermitian manifold and let \(M\) be an orientable hypersurface and \(V\) a unit normal vector field on it. Then, \(\xi=JV\) is the Reeb (or characteristic) vector field of an almost contact metric structure on \(M\). The classification of those \(M\) such that \(\xi\) is a Killing vector field is an interesting problem which has been achieved for some specific \(\overline M\). For example, the classification is completely known for \(\overline M\in \{\mathbb{C},\mathbb{C} P^n, \mathbb{C} H^n\}\) and it turns out that all these hypersurfaces \(M\) are homogeneous when they are supposed to be complete. In this paper, the authors consider the case when \(\overline M\) is the complex Grassmann manifold \(G_2 (\mathbb{C}^{m+2})\), \(m\geq 3\), of all two-dimensional linear subspaces in \(\mathbb{C}^{m+2}\). These manifolds are Riemannian symmetric spaces of rank two equipped with a Kähler and a quaternionic Kähler structure. For \(m=1\), \(G_2 (\mathbb{C}^3)\) is isometric to a \(\mathbb{C} P^2\) and for \(m=2\), \(G_2(\mathbb{C}^4)\) is isometric to the real Grassmann manifold \(G^+_2(\mathbb{R}^6)\) of oriented two-dimensional linear subspaces of \(\mathbb{R}^6\). It is proved here that the connected orientable hypersurfaces \(M\) in \(G_2(\mathbb{C}^{m+2})\) with isometric Reeb vector field are just the open parts of tubes around totally \(G_2(\mathbb{C}^{m+1})\) in \(G_2 (\mathbb{C}^{m+2})\) and when \(M\) is complete they are again homogeneous.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1837388
zbMATH Keywords: complex Grassmann manifolds
zbMATH Keywords: real hypersurfaces
zbMATH Keywords: isometric Reeb flow
zbMATH Keywords: Killing vector fields
zbMATH Keywords: tubes
zbMATH Keywords: homogeneous hypersurfaces
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
full work available at URL: s00605-001-0494-4
OpenAlex ID: W1995227286
DOI: 10.1007/S00605-001-0494-4
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description: scientific article; zbMATH DE number 1837388