Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle T²M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [Analele Stiintifice ale Universitatii Al. I. Cuza 28 (1982) 63] that in the case of a finite n-dimensional manifold M, T²M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite dimensional dynamical systems.