Riemannian foliations with parallel curvature

Abstract

There is a unique Riemannian metric g on V such that f*(g) = g\ U. Let F be the Riemannian connection on V. Then P\U = f~\F). It is natural to study the relationship between the curvature of V and the structure of the foliated manifold (M, &).In the present work we study the case of parallel curvature, that is FR = 0 where R(X, Y)Z denotes the curvature tensor of F.Let 2F be a Riemannian foliation with parallel curvature of a compact manifold M. THEOREM 1. Let M be the universal cover of M and let # be the lift

De Rham decomposition theorems for foliated manifolds

“…Thus VR = 0. Hence M fibers over a simply connected Riemannian symmetric space N with the leaves of ^ as fibers [3]. Clearly N is compact and is necessarily irreducible.…”