Classes of real time-like hypersurfaces of a Kähler manifold with \(B\)-metric
type: Item
instance of: scholarly article
title: Classes of real time-like hypersurfaces of a Kähler manifold with \(B\)-metric
zbMATH Open document ID: 1042.53055
author: Mancho Manev
published in: Journal of Geometry
publication date: 2003-10-07T00:00:00Z
review text: Let \(M\) be an almost contact manifold with a structure \((\varphi,\xi,\eta)\). A pseudo-Riemannian metric \(g\) on \(M\) is called a \(B\)-metric if it satisfies the condition \(g(\varphi X,\varphi Y)=-g(X,Y)+\eta(X)\eta(Y)\) for any vector fields \(X,Y\) on \(M\). The associated metric \(\widetilde g\) given by \(\widetilde g(X,Y)=g(x,\varphi Y)+\eta(X)\eta(Y)\) is also a \(B\)-metric on \(M\). An almost contact manifold with a \(B\)-metric is called an almost contact \(B\)-metric manifold. In [Math. Balk., New Ser. 7, No. 3--4, 261--276 (1993; Zbl 0830.53031)], \textit{G.~Ganchev, V.~Mihova}, and \textit{K.~Gribachev} classified almost contact manifolds with \(B\)-metric by considering the vector space \(\mathcal F\) of tensors of type (0,3) on a \((2n+1)\)-dimensional real vector space \(V\) satisfying the same identities as the 3-covariant tensor field \(F\) on \(M\) defined by \(F(X,Y,Z)=g((\nabla_ X\varphi)Y,Z)\), where \(\nabla\) is the Levi-Civita connection of \(g\). They obtained an orthogonal decomposition of \({\mathcal F}\) into eleven subspaces \({\mathcal F}_ i\), which are invariant under the action of the group \(G\times \{1\}\), where \(G=\mathbb {GL}(n, C)\cap {\mathbb O}(n,n)\). Thus the subspaces \({\mathcal F}_ i\) give rise to \(2^{11}\) classes of all almost contact \(B\)-metric manifolds. In [Dimiev, Stancho (ed.) et al., Perspectives of complex analysis, differential geometry and mathematical physics. Proceedings of the 5th international workshop on complex structures and vector fields, St. Konstantin, Bulgaria, September 3--9, 2000. Singapore: World Scientific, 159--170 (2001; Zbl 1002.53009)], the author introduced two types of real hypersurfaces of an almost complex \(B\)-metric manifold. The first type is a real time-like hypersurface that is defined by imposing the time-like condition on a normal unit vector \(N\), and the second type is an isotropic hypersurface with respect to the associated metric \(\tilde{g}\). In both cases, alm…
Mathematics Subject Classification ID: 53D15
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 1941623
zbMATH Keywords: almost contact manifold
zbMATH Keywords: \(B\)-metric
zbMATH Keywords: almost contact \(B\)-metric-manifold
zbMATH Keywords: Kähler manifold
zbMATH Keywords: real time-like hypersurface
reviewed by: Andrew Bucki
MaRDI profile type: Publication
full work available at URL: s00022-002-1579-5
OpenAlex ID: W1558680894
DOI: 10.1007/S00022-002-1579-5
description: scientific article; zbMATH DE number 1941623