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Weakly symmetric Riemannian manifolds with a reductive isometry group. (English. Russian original) Zbl 1078.53043
Sb. Math. 195, No. 4, 599-614 (2004); translation from Mat. Sb. 195, No. 4, 143-160 (2004).
A Riemannian manifold $$M$$ is said to be weakly symmetric if any two points in $$M$$ can be interchanged by an isometry. Clearly any symmetric space is weakly symmetric. In the present paper the author derives a classification, up to local isomorphism, of all non-symmetric weakly symmetric Riemannian manifolds with a reductive isometry group.
The main idea for the classification is as follows: A weakly symmetric Riemannian manifold with reductive isometry group is a weakly symmetric homogeneous space and therefore a real form of some spherical homogeneous space. The spherical homogeneous spaces and their real forms are classified. If a homogeneous space $$G/K$$ is a symmetric Riemannian manifold for some choice of a $$G$$-invariant Riemannian metric, then there exists a factorization $$\text{Isom}(G/K) = GQ$$, where $$Q$$ is a symmetric subgroup of $$\text{Isom}(G/K)$$. The factorizations of reductive groups are described by Onishchik, which gives a clue to understanding whether $$G/K$$ can be a symmetric Riemannian manifold. Via a sequence of reductions the problem of the classification of non-symmetric weakly symmetric Riemannian manifolds is reduced to the consideration of certain homogeneous spaces.

##### MSC:
 53C30 Differential geometry of homogeneous manifolds
##### Keywords:
weakly symmetric spaces; reductive isometry groups
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