Foundations of Physics, Vol. 24. No. 5, 1994 Two Questions on the Geometry of Gauge Fields N. C. A. da Costa,~ F. A. Doria, 2 A. F. Furtado-do-Amaral, 3 and J. A. de Barros 4 Received October 9, 1991; revised July 7, 1992 We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given con- nection form satisfies the preceding conditions. A parallel result gives a version of G6del's first incompleteness theorem within an (axiomatized) theory of gauge fields. 1. INTRODUCTION The present paper deals with two interlocked questions in theoretical and mathematical physics. We are interested in a given phenomenon; we for- malize it in a reasonable way, and state a result that the describes some "nice" necessary and sufficient conditions for that phenomenon to happen. Then, given a specific example, can we actually check whether our example satisfies the prescribed conditions? The geometrical question we deal with here started out of a simple and yet interesting problem in the mathematics of gauge fields, the problem of the "Bianchi condition copies." That problem can be stated as follows: t Institute for Advanced Studies, University of S~o Paulo, Av. Prof. Luciano Gualberto, trav. J, 374, 05655-010 S~o Paulo SP Brazil. 2 Center for the Study of Mathematical Theories of Communication, School of Communica- tions, Federal University of Rio de Janeiro, Av. Pasteur, 250. 22295-900 Rio de Janeiro RJ Brazil. 3 Institute of Physics, Federal University of Rio de Janeiro, 21949-900 Rio de Janeiro RJ Brazil. 4 Brazilian Center for Physical Research, R. Dr. Xavier Sigaud, 150. 22290-000 Rio de Janeiro RJ Brazil. 783 0015-9018/94/0500-0783507.00/0 9 1994 PlenumPublishing Corporation