YANG-MILLS CONNECTIONS ON A COMPACT CONNECTED SEMISIMPLE LIE GROUP

Abstract

Abstract. Let G be a compact connected semisimple Lie group, g theLie algebra of G, g the canonical metric (the biinvariant Riemannianmetric which is induced from the Killing form of g), and r be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection r in the tangent bundle TG over (G;g) is a Yang-Millsconnection. 1. IntroductionThe problem of nding metrics and connections which are critical pointsof some functional plays an important role in global analysis and Riemann-ian geometry. A Yang-Mills connection is a critical point of the Yang-MillsfunctionalYM(D) =12Z M kR D k 2 v g (1:1)on the space C E of all connections in a smooth vector bundle Eover a closed(compact and connected) Riemannian manifold (M;g), where R D is the cur-vature of D2C E . Equivalently, Dis a Yang-Mills connection if it satis es theYang-Mills equation (cf. [1, 5, 6]) D R D = 0; (1:2)(the Euler-Lagrange equations of the variational principle associated with (1.1)).If Dis a connection in a vector bundle Ewith bundle metric hover a Riemann-ian manifold (M;g), then the connection D given byh(D