zbMATH — the first resource for mathematics

Weakly symmetric spaces. (English) Zbl 0860.53030
Gindikin, Simon (ed.), Topics in geometry. In memory of Joseph D’Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 355-368 (1996).
A Riemannian manifold \((M,g)\) is called a weakly symmetric space if for any two points \(p\) and \(q\) in \(M\) there exists an isometry of \(M\) which interchanges \(p\) and \(q\). Symmetric spaces are trivial examples but there are many non-symmetric ones. The weakly symmetric spaces share a lot of properties with symmetric ones. For example, the geodesic symmetries are volume-preserving up to sign; each geodesic is an orbit of a one-parameter subgroup of isometries; the algebra of all isometry-invariant differential operators is commutative. But there are also differences. For example, the author shows that there are weakly symmetric spaces which are homeomorphic but not diffeomorphic.
The main purpose of this interesting paper is the construction of a rich collection of examples of weakly symmetric spaces which (together with the already known examples) show that there is now a wealth of examples which make it worthwhile to continue a deeper study of this class of spaces.
For the entire collection see [Zbl 0842.00040].

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds