Transversally symmetric Riemannian foliations (Q2639351)

A Riemannian foliation F is transversally symmetric if its transversal geometry is locally modelled on a Riemannian symmetric space. The paper under review is concerned with the study of such foliations on a Riemannian manifold (M,g) endowed with a bundle-like metric g. The authors characterize the transversal symmetry in terms of the Levi-Civita connection of the normal bundle. This characterization can be sharpened for a totally geodesic foliation F, at least in the analytic case. In particular, in such a case the transversal symmetry is characterized geometrically by the fact that the reflections with respect to the leaves are isometries. The previous results are applied to a manifold (M,g) of constant sectional curvature and to a Kaehler manifold (M,g,J) of constant holomorphic sectional curvature. In the last section, the special case of the Riemannian flow (Riemannian foliation with one- dimensional leaves) defined by a unit Killing vector field is considered.