Canonical flows of Einstein-Weyl manifolds (Q2752851)

A manifold \(M\) is a Weyl manifold if it has a conformal structure \([g]\) and a compatible torsion-free affine connection \(D\). The compatibility condition is equivalent to the existence of a 1-form \(\omega\) with \(Dg=\omega \otimes g\) for each \(g\) in the conformal structure. \(M\) is Einstein-Weyl if the symmetrized Ricci tensor of \(D\) is pointwise proportional to \(g\). It admits a strict Gauduchon gauge if one can choose a representative metric and nonsingular 1-form \(\omega\) that is dual to a Killing vector field \(\omega^{\#}\) of constant nonzero length. Given such a gauge \((g,\omega)\), the canonical flow is the flow whose leaves are the integral curves of \(\omega^{\#}\). NEWLINENEWLINENEWLINEThe author investigates conditions such that the canonical flow on a compact Einstein-Weyl manifold admits a transversal Kähler structure. In particular, he proves that if the flow is transversally Einstein, then if the constant conformal scalar curvature \(s_{D} > 0\), the flow admits a transversal Kähler structure. If \(s_{D} = 0\), then \(M\) is locally a Riemannian product with first Betti number \(b_{1}(M) = 1\). NEWLINENEWLINENEWLINEHe further constructs examples of compact Einstein-Weyl manifolds that admit strict Gauduchon gauges.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].