## Yang-Mills connections with Weyl structure.(English)Zbl 1159.53016

For an arbitrary given connection $$D$$ which is not necessary metric or torsion-free in the tangent bundle $$TM$$ over an $$n$$-dimensional closed Riemannian manifold $$(M,g)$$, the following main results are proved.
Theorem 1. Let $$(M,g)$$ be a closed Riemannian manifold and $$(D,g,\omega)$$ a Weyl structure in the tangent bundle $$TM$$ over $$(M,g)$$. Then
$(\delta_{D^*}R^{D^*}-\delta_DR^D)(X)Y=(\delta\text{d}\omega)(X)Y,\quad (X\in\setminus \mathfrak X(M),\quad Y\in \Gamma(E)).$
Here, $$\mathfrak X(M)$$ denotes the space of vector fields on $$M$$.
Corollary. If $$D$$ is a Yang-Mills connection with Weyl structure $$(D,g,\omega)$$ in the tangent bundle $$TM$$ over a closed Riemannian manifold $$(M,g)$$, then $$\text{ d}\omega=0$$.
Theorem 2. Let $$D$$ be a connection with Weyl structure $$(D,g,\omega)$$ in the tangent bundle over a closed Riemannian manifold $$(M,g)$$, and $$\nabla$$ the Levi-Civita connection of $$(M,g)$$. Assume $$\sum_{i=1}^n[\alpha(e_i),R^D(e_i,X)]=0$$, where $$X\in\setminus \mathfrak X(M)$$ and $$D-\nabla=\alpha$$ and $$\{e_i\}^n_{i=1}$$ is an (locally defined) orthonormal frame on $$(M,g)$$. Then the following statements are equivalent:
(i) $$D$$ is a Yang-Mills connection.
(ii) $$\delta_{\nabla}R^D=0$$.

### MSC:

 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)

### Keywords:

Yang-Mills connection; conjugate connection; Weyl structure
Full Text:

### References:

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