In this paper we present original variational formulations of Yang–Mills, Einstein’s gravitation and Kaluza–Klein theories, where, in the spirit of General Relativity, the principal bundle structure over the space-time is not fixed a priori but is dynamical. In the Yang–Mills case only a topological fibration is given a priori. In the gravity and the Kaluza–Klein theories no fibration is assumed: any critical point of the action functional defines a foliation of the manifold and the leaves make up the space-time. The latter is naturally equipped with a pseudo-Riemannian metric and, under some hypotheses, this foliation is actually a fibration. In all cases the apparition of a (at least local) principal bundle structure and a connection follows from the dynamics. Moreover the metric and the connection thus constructed are solutions of the Yang–Mills, the Einstein–Cartan or the Yang–Mills–Einstein equations, depending on the model. A crucial point is that we face the difficulty that some Lagrange multiplier fields (which are responsible for the foliation, the principal bundle structure and the connection) create unwanted terms in the equations. This difficulty is overcome by the observation that, if the structure group is compact, these terms miraculously cancel.
Frédéric Hélein (Tue,) studied this question.