YANG-MILLS INDUCED CONNECTIONS

Let G and H be compact connected Lie groups with biinvariant Riemannian metrics g and h respectively, Á a group isomorphism of G onto H, and E := Á¡1TH the induced bundle by Á over the base manifold G of the tangent bundle TH of H. Let r and Hr be the Levi-Civita connections for the metrics g and h respectively, ~r the induced connection by the map Á and Hr. Then, a necessary and su±cient condition for ~r in the bundle (Á¡1TH; G; ¼) to be a Yang- Mills connection is the fact that the Levi-Civita connection r in the tangent bundle over (G; g) is a Yang- Mills connection. As an application, we get the following: Let à be an automorphism of a compact connected semisimple Lie group G with the canonical metric g (the metric which is induced by the Killing form of the Lie algebra of G), r the Levi-Civita connection for g. Then, the induced connection ~r, by à and r, is a Yang-Mills connection in the bundle (Á¡1TG; G; ¼) over the base manifold (G; g).