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.

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.

Reviewer: Jürgen Berndt (Cork)

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |