zbMATH — the first resource for mathematics

Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. (English) Zbl 0631.53033
Differential geometry on homogeneous spaces, Conf. Torino/Italy 1983, Rend. Semin. Mat., Torino, Fasc. Spec., 131-158 (1983).
[For the entire collection see Zbl 0624.00013.]
An explicit classification of all 3-dimensional (simply connected and complete) Riemannian manifolds with volume-preserving local geodesic symmetries is given. Some related classes of Riemannian manifolds are discussed in this context, namely the harmonic spaces, the naturally reductive spaces, the Riemannian homogeneous spaces on which all invariant differential operators commute, and the Riemannian manifolds on which every two random steps commute.

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