Geometry of the double tangent bundles of Banach manifolds

In this paper for a vector bundle (v.b.) (p,E,M)(p,E,M), we show that at the presence of a (possibly nonlinear) connection on (p,E.M)(p,E.M), TETE on MM admits a v.b. structure. This fact is followed by a suitable converse which asserts that a v.b. structure for TETE over MM yields a linear connection on the original bundle (p,E,M)(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 MM is introduced which leads us to interesting geometric tools on the bundle of accelerations  . In fact by using the v.b. structure for σ:TTM⟶Mσ: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.