Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

Abstract

The main results of this paper are as follows. (a) Let π : M → N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kahler structure andM a Sasakian structure. (b) Let \( \pi:{M^{2n + 1} \to N^{2n}} \) be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature \( \geqslant 4n(n + 1)\). If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.