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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)
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References:

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