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New examples of Riemannian g.o. manifolds in dimension 7. (English) Zbl 1050.22011

Authors’ abstract: A Riemannian g.o. manifold is a homogeneous Riemannian manifold \((M,g)\) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension \(\leq 5\) is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian manifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.

MSC:

22E25 Nilpotent and solvable Lie groups
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
53C40 Global submanifolds
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