Let G and H be compact connected Lie groups with biinvariant Riemannian metrics g and h respectively,

a group isomorphism of G onto H, and E :=

TH the induced bundle by

over the base manifold G of the tangent bundle TH of H. Let

and

be the Levi-Civita connections for the metrics g and h respectively,

the induced connection by the map

and

. Then, a necessary and sufficient condition for

in the bundle (

) to be a Yang- Mills connection is the fact that the Levi-Civita connection

in the tangent bundle over (G, g) is a Yang- Mills connection. As an application, we get the following: Let

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),

the Levi-Civita connection for g. Then, the induced connection

, by

and

, is a Yang-Mills connection in the bundle (

) over the base manifold (G, g).