Cohomogeneity one actions on hyperbolic spaces. (English) Zbl 1014.53042
Authors’ summary: We present a new method for constructing cohomogeneity one actions on the hyperbolic spaces over the complex, quaternionic and Cayley numbers. As applications we obtain negative answers to the following two open problems: (1) Is the moduli space of all cohomogeneity one actions (modulo orbit equivalence) on a simply connected symmetric space always a finite set? The answer was known to be yes for symmetric spaces of compact type, for Euclidean spaces, and for real hyperbolic spaces. (2) Is a singular orbit of a cohomogeneity one action on a Hadamard manifold always totally geodesic? The answer was known to be yes for Euclidean spaces and real hyperbolic spaces. We also classify all cohomogeneity one actions with a singular totally geodesic orbit on these hyperbolic spaces, and give a detailed description of the new examples. The discussion in the complex case is based on the Kähler angle, in the quaternionic case it is based on a new geometric quantity which we call quaternionic Kähler angle, and in the Cayley case it is based on the spin representation of Spin(7).
Reviewer: W.Mozgawa (Lublin)
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53C35 | Differential geometry of symmetric spaces |
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