YANG-MILLS INDUCED CONNECTIONS

Abstract

Let G and H be compact connected Lie groups with biinvariant Riemannian metrics g and h respectively, φ a group isomorphism of G onto H, and E := φ−1T H the induced bundle by φ over the base manifold G of the tangent bundle T H of H. Let ∇ and H ∇ be the Levi-Civita connections for the metrics g and h respectively, ˜∇ the induced connection by the map φ and H ∇. Then, a necessary and sufficient condition for ˜∇ in the bundle (φ−1T H, G, π) to be a YangMills connection is the fact that the Levi-Civita connection ∇ in the tangent bundle over (G, g) is a YangMills connection. As an application, we get the following: Let ψ 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 ψ and ∇, is a Yang-Mills connection in the bundle (φ−1T G, G, π) over the base manifold (G, g).

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