WEYL STRUCTURES ON COMPACT CONNECTED LIE GROUPS

Abstract

Let G be a compact connected semisimple Lie group, B the Killing form of the algebra g of G, and g the invariant metric induced by B. Then, we obtain a necessary and sufficient condition for a left invariant linear connection D with a Weyl structure (D, g, ω) on (G, g) to be projectively flat (resp. Einstein-Weyl). And, we also get that if a left invariant linear connection D with a Weyl structure (D, g, ω) on (G, g) which has symmetric Ricci tensor RicD is projectively flat, then the connection D is EinsteinWeyl; but the converse is not true. Moreover, we show that if a left invariant connection D with Weyl structure (D, g, ω) on (G, g) is projectively flat (resp. Einstein-Weyl), then D is a Yang-Mills connection.

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