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By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we give, for any compact connected semisimple Lie group G, a general procedure to obtain the continuous fibrations of G by Weyl-oriented affine maximal tori, find conditions for smoothness and provide infinite dimensional spaces of concrete examples. 2000 Mathematics Subject Classification: 55R15, 22E15, 53C12 * Partially supported by conicor, ciem (conicet) and secyt (unc)doi:10.1142/s0129167x02001216 fatcat:qubzpcdmzjcdbl25k25rgrroc4