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July 2008 Yang-Mills connections with Weyl structure
Joon-Sik Park
Proc. Japan Acad. Ser. A Math. Sci. 84(7): 129-132 (July 2008). DOI: 10.3792/pjaa.84.129


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 $(M,g)$.


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Joon-Sik Park. "Yang-Mills connections with Weyl structure." Proc. Japan Acad. Ser. A Math. Sci. 84 (7) 129 - 132, July 2008.


Published: July 2008
First available in Project Euclid: 17 July 2008

zbMATH: 1159.53016
MathSciNet: MR2450065
Digital Object Identifier: 10.3792/pjaa.84.129

Primary: 53A15 , 53C07

Keywords: conjugate connection , Weyl structure , Yang-Mills connection

Rights: Copyright © 2008 The Japan Academy

Vol.84 • No. 7 • July 2008
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