Symmetries on compact pseudo-Hermitian manifolds
type: Item
instance of: scholarly article
title: Symmetries on compact pseudo-Hermitian manifolds
zbMATH Open document ID: 0652.53020
DOI: 10.1007/BF02844708
author: Giuliana Gigante
publication date: 1987-01-01T00:00:00Z
review text: A pseudo-Hermitian manifold \((M^*,w)\) consists of a \(C^{\infty}\) manifold \(M^*\) of dimension \(2n+1\) and of a global real 1-form w for which there are locally 2n complex 1-forms \(w_ 1,...,w_ n\), \(w_{\bar 1},...,w_{\bar n}\) with \(w_{\bar i}=\bar w_ i\), that together with w determine a local basis for the cotangent space of \(M^*\) and such that \(dw=-i g_{\alpha {\bar \beta}}w_{\alpha}\wedge w_{{\bar \beta}},\) where \(g=(g_{\alpha {\bar \beta}})\) is a nondegenerate Hermitian matrix; if it is negative definite, then \((M^*,w)\) is said to be strongly pseudoconvex. In this paper the author proves the following theorem: Any compact, simply connected, pseudo- Hermitian strongly pseudoconvex, symmetric manifold \((M^*,w)\) is a real hypersurface of a holomorphic line bundle F over a symmetric Hermitian manifold, whose Kähler form is, up to a multiplicative constant, the Chern class of F. More specifically \((M^*,w)\) is an \(S^ 1\)-principal bundle associated to F. It would be interesting to know whether the condition that \((M^*,w)\) has to be strongly pseudoconvex may be dropped.
Mathematics Subject Classification ID: 53B35
zbMATH DE Number: 4063968
zbMATH Keywords: pseudo-Hermitian manifold
zbMATH Keywords: strongly pseudoconvex
zbMATH Keywords: hypersurface
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: Kähler form
reviewed by: Cheng-Chung Hwang
MaRDI profile type: Publication
cites work: Q5531609
cites work: Real hypersurfaces in complex manifolds
cites work: Q3846750
cites work: Q5565587
cites work: Q4089110
cites work: Q5825483
cites work: Pseudo-hermitian structures on a real hypersurface
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4063968