Compact Einstein-Weyl manifolds and the associated constant

A Weyl structure on a manifold M is a torsion free affine connection D preserving a conformal structure [g]. Namely, a torsion free affine connection D is called a Weyl structure if Dg — ω ® g for a 1-form ω. The definition of Weyl structure goes back to the work of H. Weyl. In his famous book ([23]) he introduced Weyl structure to unify gravitational fields and electromagnetic fields. The notion of Einstein-Weyl structure is originated in the paper of N.Hitchin ([11]) in which he developed the 3-dimensional minitwistor theory associated to the 3-dimensional monopole theory and observed that the minitwistor theory can be generalized over a 3-manifold endowed with a Weyl structure obeying a certain Ricci tensor condition, namely an Einstein-Weyl structure. Refer also to [12]. The exact definition of Einstein-Weyl structure is the following. A Weyl structure (-D, [g]) is Einstein-Weyl if the symmetrized Ricci tensor is proportional to a metric g representing [#];