Real hypersurfaces in Hermitian symmetric spaces
The main purpose of this monograph is the description of real hypersurfaces in Hermitian symmetric spaces such as: the complex quadric \(\mathcal{Q}^n=\mathrm{SO}_{n+2}/\mathrm{SO}_n\mathrm{SO}_2\) for \(n\geq 3\), the complex hyperbolic quadric \(\mathcal{Q}^{n*}=\mathrm{SO}_{n,2}/\mathrm{SO}_n\) for \(n\geq 3\), the complex 2-plane Grassmannian \(G_2(\mathbb C^{k+2})=\mathrm{SU}_{k+2}/\mathrm{S}(\mathrm{U}_k \mathrm{U}_2)\) for \(k\geq 3\) and the complex hyperbolic 2-plane Grassmannian \(G_2^*(\mathbb C^{k+2})=\mathrm{SU}_{k,2}/\mathrm{S}(\mathrm{U}_k\mathrm{U}_2)\) for \(k\geq 3\). The basic motivation is the attempt to generalize the classical theory of real hypersurfaces in complex projective spaces and complex hyperbolic spaces to other Kähler manifolds. Many results described in this monograph have been previously obtained by the authors and their collaborators, result of many years of collaborative research, but other material is enclosed. The book presents eight chapters and six pages of bibliography. Even if the reader is expected to have a basic knowledge of Riemannian geometry and of the theory of submanifolds, in the first chapter the authors recap basic concepts from Riemannian geometry, Kähler geometry, Riemannian and Hermitian symmetric spaces and, in the second chapter, the basic concepts of submanifolds theory are presented: fundamental equations, totally geodesic submanifolds, curvature-adapted submanifolds, focal sets, tubes, homogeneous hypersurfaces. Chapter 3 is devoted to the study of real hypersurfaces in Kähler manifolds, in particular: Hopf hypersurfaces, hypersurfaces with isometric Reeb flow, contact and hypersurfaces in complex space forms. The main purpose of the remaining four chapters is the description of the geometry of real hypersurfaces in complex 2-plane Grassmannians, in complex hyperbolic 2-plane Grassmannians, in complex quadrics and in complex hyperbolic quadrics. Finally, Chapter 8 is devoted to the generalization to Hermitian symmetric spaces of compact type. The book is densily written, various parts are in the style of a review, with comments and references, however most theorems and statements have a proof. Concluding, this is a well-written, interesting book, describing the state of the art on the considered topic.
scientific article; zbMATH DE number 7438184