Weakly symmetric groups of Heisenberg type
A weakly symmetric space is a Riemannian manifold on which any two points can be interchanged by an isometry. This class of Riemannian homogeneous spaces was introduced by \textit{A. Selberg} [J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)]. In the present paper, the authors classify all generalized Heisenberg groups which are weakly symmetric. It turns out that these are precisely those generalized Heisenberg groups which are commutative spaces. A commutative space is a Riemannian manifold on which the algebra of all isometry-invariant differential operators is commutative. The commutative generalized Heisenberg groups have been classified previously by the second author in [\textit{F. Ricci}, J. London Math. Soc. 32, 265-271 (1985; Zbl 0592.22009)]. The authors also classify all generalized Heisenberg groups on which any two points can be interchanged by an isometry belonging to the identity component of the full isometry group. Some weakly symmetric generalized Heisenberg groups turn out not to have this property. The authors further provide a brief survey about weakly symmetric spaces.
scientific article; zbMATH DE number 1186364
scientific article