WEYL STRUCTURES ON COMPACT CONNECTED LIE GROUPS

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 su±cient condi- tion for a left invariant linear connection D with a Weyl structure(D; g; !) on (G; g) to be projectively °at (resp. Einstein-Weyl). And, we also get that if a left invariant linear connection D with a Weyl structure (D; g; !) on (G; g) which has symmetric Ricci ten-sor RicD is projectively °at, then the connection D is Einstein- Weyl; but the converse is not true. Moreover, we show that if a left invariant connection D with Weyl structure (D; g; !) on (G; g) is projectively °at (resp. Einstein-Weyl), then D is a Yang-Mills connection.