Pak, Jin Suk; Kim, Hyang Sook Application of an integral formula to CR-submanifolds of complex hyperbolic space. (English) Zbl 1084.32014 Int. J. Math. Math. Sci. 2005, No. 7, 987-996 (2005). The authors use an integral formula established by K. Yano [Integral formulas in Riemannian geometry (Pure and Applied Mathematics. New York: Marcel Dekker) (1970; Zbl 0213.23801)] to investigate compact CR submanifolds \(M\) of hypersurface type of the complex hyperbolic space \(X=\mathbb{C} H^m\) of constant holomorphic sectional negative curvature \(c=-4\). They show (Theorem 3.3) that, under suitable conditions on the normal vector field to the CR distribution and on the Ricci and scalar curvature, the CR manifold \(M\) has a generic embedding into a totally geodesic submanifold \(Y\) of \(X\), and that, as a real submanifold of \(Y\), its almost contact structure \(F\) and its second fundamental form \(A\) satisfy the commutativity condition \(AF=FA\). Finally, they classify the generic CR hypersurfaces of \(X\) satisfying \(AF=FA\), and show that the inequality used in Theorem 3.3. can be used to characterize the geodesic hyperspheres of \(X\). Reviewer: Mauro Nacinovich (Roma) Cited in 2 Documents MSC: 32Q05 Negative curvature complex manifolds 32Q57 Classification theorems for complex manifolds 32V30 Embeddings of CR manifolds Keywords:Hyperbolic space; CR submanifold Citations:Zbl 0213.23801 PDFBibTeX XMLCite \textit{J. S. Pak} and \textit{H. S. Kim}, Int. J. Math. Math. Sci. 2005, No. 7, 987--996 (2005; Zbl 1084.32014) Full Text: DOI EuDML