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Cohomogeneity one actions on noncompact symmetric spaces of rank one. (English) Zbl 1117.53041

The authors of the present paper classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic, and Cayley numbers, and on the complex hyperbolic spaces \(\mathbb{C} H^n\), \(n\geq 3\). For the quaternionic hyperbolic spaces \(\mathbb{H} H^n\), \(n\geq 3\), the authors reduce the classification problem to a problem in the quaternionic linear algebra and obtain partial results. It is worth being mentioned here that for real hyperbolic spaces, this classification problem was essentially solved by Cartan.

MSC:

53C35 Differential geometry of symmetric spaces
57S20 Noncompact Lie groups of transformations
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