In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.
A new methodology leading to the construction of a universal connection for Frchet principal bundles is proposed in this
paper. The classical theory, applied successfully so far for finite dimensional and Banach modelled bundles, collapses within
the framework of Frchet manifolds. However, based on the replacement of the space of continuous linear mappings by an appropriate
topological vector space, we endow the bundle J1P of 1-jets of the sections of a Frchet principal bundle P with a connection form by means of which we may “reproduce” every connection of P.