Real hypersurfaces of a Kähler manifold (the sixteen classes) (Q2765982)

Let \(\overline M\) be a \(2n\)-dimensional almost Hermitian manifold with an almost complex structure \((J,G)\) and let \(F_J(X,Y,Z)=G((\nabla_XJ)Y,Z)\). In [Ann. Mat. Pura Appl., IV. Ser. 123, 35-58 (1980; Zbl 0444.53032)] \textit{A. Gray} and \textit{L. M. Hervella} have classified almost Hermitian manifolds by obtaining characterizations of sixteen different classes. They considered the vector space \(\mathcal W_J\) of tensors of type (0,3) of a \(2n\)-dimensional vector space \(V\) with the same symmetry properties as the tensor \(F_J\) with respect to the Levi-Civita connection \(\nabla\) associated to \(G\) and studied the decomposition of this space as a direct sum of subspaces invariant and irreducible under the natural action of the unitary group \({\text U}(n, R)\) on \(\mathcal W_J\). They found four terms \(W_i\;(i=1,2,3,4)\) in this decomposition. NEWLINENEWLINENEWLINELet \(M\) be a \((2n-1)\)-dimensional manifold with an almost contact metric structure \(\Sigma=(\varphi,\) \(\xi,\eta, g)\) and let \(F_\varphi\) be a 3-covariant tensor field on \(M\) defined by \(F_\varphi(X,Y,Z)=-g((\nabla_X\varphi)Y,Z)\). In [Mathematics and education in mathematics, Proc. 15th Spring Conf., Sunny Beach/Bulg. 186-191 (1986; Zbl 0601.53031)] the first author and \textit{V. A. Aleksiev} classified almost contact metric manifolds by considering the vector space \(\mathcal W_\varphi\) of tensors of type (0,3) on a \((2n-1)\)-dimensional real vector space \(V\) satisfying the same identities as \(F_\varphi\). They obtained 12 natural basic invariant subspaces of \(\mathcal W_\varphi\) as a decomposition of \(\mathcal W_\varphi\) into orthogonal components invariant under the action of \({\text U}(n,\mathbb{R})\times 1\). Thus they obtained twelve basic classes \(W_i\;(i=1,2,\dots,12)\) of almost contact metric manifolds. For example, the subspace \(W_6\oplus W_{11}\) is the class of almost cosymplectic manifolds and \(W_2\oplus W_3\oplus W_4\oplus W_5\oplus W_9\oplus W_{10}\) is the class of normal almost contact metric manifolds.NEWLINENEWLINENEWLINEIn this paper, the authors study properties of real hypersurfaces of Kähler manifolds. For a \(2n\)-dimensional almost Hermitian manifold \(\overline M\) with an almost complex structure \((J,G)\) let \(M\) be its real hypersurface with a unit normal vector field \(N\). Then the structure \((\varphi,\xi,\eta, g)\), defined by \(\xi=-JN\), \(g=G_{|M}\), \(\eta(X)=g(X,\xi)\), and \(\varphi=J-\eta\otimes N\), is an almost contact metric structure on \(M\). The authors prove that any real hypersurface \(M\) with the induced almost contact metric structure \((\varphi,\xi,\eta, g)\) of a Kähler manifold \(\overline M\) with a structure \((J,G)\) is in the class \(W_1\oplus W_2\oplus W_4\oplus W_6\). The four basic classes \(W_1\), \(W_2\), \(W_4\), and \(W_6\) generate sixteen classes of real hypersurfaces. The examples of some geometric conditions characterizing real hypersurfaces in the classes \(W_1\oplus W_6\) and \(W_2\oplus W_4\) are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00014].