Generalized Hopf manifolds, locally conformal Kaehler structures and real hypersurfaces
First the author studies a certain action of \(\mathbb{Z}\) on the product of the punctured open unit disk in \(\mathbb{C}\) and some Brieskorn manifold. The resulting quotient space is proved to be a complex manifold and, in dimension two, a complex analytic family of compact complex surfaces. Then he turns to the complex Hopf manifold \(H^ n_ a\), which is the quotient space of \(\mathbb{C}^ n\setminus\{0\}\) by the group of transformations of \(\mathbb{C}^ n\setminus\{0\}\) generated by \(z\mapsto az\), \(a\in\mathbb{C}\), \(0<| a| < 1\), \(n>1\). It is known, that \(H^ n_ a\) admits no globally defined Kähler metric, and the author proves that this also holds for the complex sphere \(\{z\in H^ n_ a\mid z^ 2_ 1+\dots+z^ 2_ n=0\}\) in \(H^ n_ a\), provided that \(a\in\mathbb{R}\) and \(n>4\). Next he shows that the canonical projection \(H^ n_ a\to \mathbb{C} P^{n-1}\) becomes a Riemannian submersion with minimal fibers, when endowing \(H^ n_ a\) with the Boothby metric and \(\mathbb{C} P^{n-1}\) with the Fubini-Study metric. A \(P_ 0K\)-manifold is a locally conformal Kähler manifold with parallel Lee form whose local Kähler metrics are flat. According to \textit{I. Vaisman} [Rend. Mat., VI. Ser. 12, 263-264 (1979; Zbl 0447.53032)] any \(P_ 0K\)-manifold is locally analytically homothetic to some \(H^ n_ a\) with the Boothby metric. In the latter part of the paper the author derives several results about real hypersurfaces with two principal curvatures, quasi-Einstein real hypersurfaces and homogeneous real hypersurfaces in \(P_ 0K\)-manifolds. Finally he proves that every totally umbilical submanifold in \(H^ n_ a\) tangent to the Lee field is either totally geodesic or an extrinsic sphere.
scientific article; zbMATH DE number 55872
scientific article