Yang–Mills theory and conjugate connections

✍ Scribed by Sorin Dragomir; Toshiyuki Ichiyama; Hajime Urakawa

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 101 KB
• Volume: 18
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(02)00149-3

✦ Synopsis

We develop a new Yang-Mills theory for connections D in a vector bundle E with bundle metric h, over a Riemannian manifold by dropping the customary assumption Dh = 0. We apply this theory to Einstein-Weyl geometry (cf. M.F. Atiyah, et al., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London 362 (1978) 425-461, and H. Pedersen, et al., Einstein-Weyl deformations and submanifolds, Internat. J. Math. 7 (1996) 705-719) and to affine differential geometry (cf. F. Dillen, et al., Conjugate connections and Radon's theorem in affine differential geometry, Monatshefts für Mathematik 109 (1990) 221-235). We show that a Weyl structure (D, g) on a 4-dimensional manifold is a minimizer of the functional (D, g) → 1 2 M R D 2 v g if and only if * R D = ±R D * , where D * is conjugate to D. Moreover, we show that the induced connection on an affine hypersphere M is a Yang-Mills connection if and only if M is a quadratic affine hypersurface.