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Application of an integral formula to CR-submanifolds of complex hyperbolic space. (English) Zbl 1084.32014

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 \(X\).

MSC:

32Q05 Negative curvature complex manifolds
32Q57 Classification theorems for complex manifolds
32V30 Embeddings of CR manifolds

Citations:

Zbl 0213.23801
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