Yang-Mills connections with Weyl structure

Abstract

In this paper, we treat with an arbitrary given connection $D$ which is not necessarily \textit{metric} or \textit{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 $\sum_{i=1}^{n}[\alpha(e_{i}),R^{D}(e_{i},X)]=0$ $(X \in \mathfrak{X}(M))$, a necessary and sufficient condition for any connection $D$ with Weyl structure $(D,g,\omega)$ to be a Yang-Mills connection is $\delta_{\nabla}R^{D}=0$, where $\{e_{i}\}_{i=1}^{n}$ is an (locally defined) orthonormal frame on $(M,g)$ and $D-\nabla = \alpha \in \Gamma (\bigwedge TM^{\ast} \otimes \mathrm{End}(TM))$, and $\nabla$ is the Levi-Civita connection for $g$ of $(M,g)$.