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Geometry of weakly symmetric spaces. (English) Zbl 0877.53027
Weakly symmetric spaces have been introduced by A. Selberg in [J. Indian Math. Soc., n. Ser. 20, 47-87 (1956; Zbl 0072.08201)]. An alternative definition says that a Riemannian manifold \(M\) is weakly symmetric if, for any two points \(p,q\) in \(M\), there exists an isometry of \(M\) interchanging \(p\) with \(q\). The present authors give many examples of weakly symmetric spaces which are not symmetric. First, the geodesic tubes in \(\mathbb{C} P^n\) around points or naturally embedded \(\mathbb{C} P^1, \dots, \mathbb{C} P^{n-1}\) are weakly symmetric spaces. The analogues hold for the quaternionic space \(\mathbb{H} P^n\), the Cayley plane \(\text{Cay} P^2\) and for the noncompact duals \(\mathbb{C} H^n\), \(\mathbb{H} H^n\), \(\text{Cay} H^2\) (in which the tubes are horospheres). Finally, the full classification of all simply connected weakly symmetric spaces in dimensions three and four is given. One can see that these coincide with the naturally reductive spaces. (The reviewer and R. A. Marinosci extended this classification to dimension five using a more algebraic method [O. Kowalski and R. A. Marinoski, J. Geom. 58, 123-131 (1997; Zbl 0868.53038)]).
Reviewer: O.Kowalski (Praha)

53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
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