Yang-Mills connections with Weyl structure

In this paper, we treat with an arbitrary given connection D which is not necessarily metric or torsion-free in the tangent bundle TM over an n-dimensional closed (compact and connected) Riemannian manifold (M, g). We find the fact that if any connection D with Weyl structure (D,g,omega) relative to a 1-form omega in the tangent bundle is a Yang-Mills connection, then d omega = 0. Moreover under the assumption Sigma(n)(i=1) [alpha(e(i)), R-D(e(i), X)] = 0 (X is an element of x (M)), a necessary and sufficient condition for any connection D with Weyl structure (D, g, omega) to be a Yang-Mills connection is delta R-del(D) = 0 where {e(i)}(i=1)(n) is an (locally defined) orthonormal frame on (M, g) and D - del= alpha is an element of Gamma(boolean AND TM* circle times End (T M)), and del is the Levi-Civita connection for g of (M, g).