Let M be a manifold with a conformal structure [g] and a tortion-free affine connection D.
I consider a Gauduchon manifold (M, g, D) with the Gauduchon metric g. Firstly, I obtained that a Gauduchon manifold (M, g, D) (n【greater than or equal】4) is a Gauduchon manifold of constant curvature if and only if (M, g, D) is a Weyl conformally flat Einstein-Weyl manifold. Next, I classified a Gauduchon manifold of constant curvature with Killing vector field. By using this result, I obtained the following result : Let (M, [g], D) be a compact Einstein-Weil manifold and Weyl conformally flat for every g【reverse surface chemistry arrow】[g]. If dimension n of M n【greater than or equal】4 and ω≠0 for every g【reverse surface chemistry arrow】[g], then (M, [g], D) is a Weyl flat manifold. Next, I investigated a Weyl submanifold of a Gauduchon manifold. Let (M, g, D) be a compact Weyl totally umbilical submanifold of a Gauduchon flat manifold which tangent to the Killing vector field B and ω≠0. Then M is a totally geodesic submanifold with Einstein-Weyl structure and the universal covering manifold of M is isometric to the Riemannian product of the sphere and R. If (M, g, D) is a Weyl hypersurface of a Gauduchon flat manifold which orthogonal to the Killing vector field B, then M is Weyl totally umbilical and a totally geodesic submanifold, moreover M is an elliptic space form. Finally, I obtained that (M, [g], J, D) admits an Einstein-Weyl structure if and only if (M, g, J) admits an Einstein-Hermitian structure.