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Naturally reductive homogeneous spaces and generalized Heisenberg groups. (English) Zbl 0551.53028

One goal of this paper is to give some applications of a theorem by Ambrose and Singer characterizing the homogeneity of a Riemannian manifold through the existence of a certain tensor field. The main application is an alternative proof of A. Kaplan’s theorem classifying all generalized Heisenberg groups which are naturally reductive. The second main topic is the detailed study of the 6-dimensional generalized Heisenberg group (the first known example of a Riemannian manifold which is in no way naturally reductive but its geodesics are still orbits of one-parameter groups of isometries).
Reviewer: O.Kowalski

MSC:

53C30 Differential geometry of homogeneous manifolds
22E25 Nilpotent and solvable Lie groups
53C20 Global Riemannian geometry, including pinching
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References:

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