First we present a unified theory of connections on bundles necessary for the next studies. For a smooth manifold M, modeled on the Banach space B, we define the bundle of linear frames LM and we endow it with a differentiable structure. Bundle of sprays FM, the pullback of LM via the tangent bundle : TM ! M, is a natural bundle which provides us a rich environment to study the geometry of M. Afterward, despite of natural difficulties with Fréchet manifolds and even spaces, we generalize these results to a wide class of Fréchet manifolds, those which can be considered as projective limits of Banach manifolds. As an alternative approach we use pre-Finsler connections on FM and we show that our technique successfully solves ordinary differential equations on these manifolds. As some applications of our results we apply our method to enrich the geometry of two known Fréchet manifolds, i.e. jet of infinite sections and manifold of smooth maps, and we provide a suitable framework for further studies in these areas.
