Classes of real time-like hypersurfaces of a Kähler manifold with \(B\)-metric
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, almost contact \(B\)-metric structures are defined on the hypersurface. It was shown that the hypersurfaces belong to the class \({\mathcal F}_ 4\oplus{\mathcal F}_ 5\oplus\cdots\oplus {\mathcal F}_ 9\oplus{\mathcal F}_ {11}\). In this paper, the author shows that the class \({\mathcal F}_ 4\oplus{\mathcal F}_ 5\oplus {\mathcal F}_ 6\oplus{\mathcal F}_ {11}\) is the class of the real time-like hypersurfaces of a Kähler manifold with \(B\)-metric, and that there are sixteen classes of these hypersurfaces and they are described by means of the second fundamental form. Also, the author constructs an example of a 3-dimensional manifold of the class \({\mathcal F}_ {11}\) as a hypersurface of the considered type of a holomorphic sphere in 6-dimensional real space.
scientific article; zbMATH DE number 1941623