Geometry of the double tangent bundles of Banach manifolds

In this paper for a vector bundle (v.b.) (p. E, M), we show that at the presence of a (possibly nonlinear) connection on (p, EM), TE on M admits a v.b. structure. This fact is followed by a suitable converse which asserts that a v.b. structure for TE over M yields a linear connection on the original bundle (p, E. M). Moreover we clarify the relation between v.b. structures and also the induced bundle morphisms which will be used for classification of these v.b. structures. Afterwards the concept of second order connections on a manifold M is introduced which leads us to interesting geometric tools on the bundle of accelerations. In fact by using the v.b. structure for sigma : TTM -> M, we will study the geometric tools on the second order tangent bundle. The concepts of second order covariant derivative, first and second order auto-parallel curve, the appropriate exponential mapping and second order Lie derivative are introduced. (C) 2013 Elsevier B.V. All rights reserved.