Proceedings of the Japan Academy, Series A, Mathematical Sciences

Yang-Mills connections with Weyl structure

Joon-Sik Park

Full-text: Open access

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 $(M,g)$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 7 (2008), 129-132.

Dates
First available in Project Euclid: 17 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1216308255

Digital Object Identifier
doi:10.3792/pjaa.84.129

Mathematical Reviews number (MathSciNet)
MR2450065

Zentralblatt MATH identifier
1159.53016

Subjects
Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20] 53A15: Affine differential geometry

Keywords
Yang-Mills connection conjugate connection Weyl structure

Citation

Park, Joon-Sik. Yang-Mills connections with Weyl structure. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 129--132. doi:10.3792/pjaa.84.129. https://projecteuclid.org/euclid.pja/1216308255


Export citation

References

  • F. Dillen, K. Nomizu and L. Vranken, Conjugate connections and Radon's theorem in affine differential geometry, Monatsh. Math. 109 (1990), no. 3, 221–235.
  • S. Dragomir, T. Ichiyama and H. Urakawa, Yang-Mills theory and conjugate connections, Differential Geom. Appl. 18 (2003), no. 2, 229–238.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978.
  • M. Itoh, Compact Einstein-Weyl manifolds and the associated constant, Osaka J. Math. 35 (1998), no. 3, 567–578.
  • A. B. Madsen et al., Compact Einstein-Weyl manifolds with large symmetry group, Duke Math. J. 88 (1997), no. 3, 407–434.
  • K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge Univ. Press, Cambridge, 1994.
  • J.-S. Park, Yang-Mills connections in the orthonormal frame bunedles over Einstenin normal homogeneous manifolds, Int. J. Pure Appl. Math. 5 (2003), no. 2, 213–223.
  • J.-S. Park, Critical homogeneous metrics on the Heisenberg manifold, Interdiscip. Inform. Sci. 11 (2005), no. 1, 31–34.
  • J.-S. Park, The conjugate connection of a Yang-Mills connection, Kyushu J. Math. 62 (2008), no. 1, 217–220.
  • H. Pedersen, Y. S. Poon and A. Swann, The Hitchin-Thorpe inequality for Einstein-Weyl manifolds, Bull. London Math. Soc. 26 (1994), no. 2, 191–194.
  • H. Pedersen, Y. S. Poon and A. Swann, Einstein-Weyl deformations and submanifolds, Internat. J. Math. 7 (1996), no. 5, 705–719.
  • H. Pedersen and A. Swann, Riemannian submersions, four-manifolds and Einstein-Weyl geometry, Proc. London Math. Soc. (3) 66 (1993), no. 2, 381–399.
  • H. Pedersen and A. Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew. Math. 441 (1993), 99–113.
  • H. Pedersen and K. P. Tod, Three-dimensional Einstein-Weyl geometry, Adv. Math. 97 (1993), no. 1, 74–109.
  • K. P. Tod, Compact 3-dimensional Einstein-Weyl structures, J. London Math. Soc. (2) 45 (1992), no. 2, 341–351.
  • H. Urakawa, Yang-Mills theory in Einstein-Weyl geometry and affine differential geometry, Rev. Bull. Calcutta Math. Soc. 10 (2002), no. 1, 7–18.