Some recent works in higher order geometry

Some recent works in higher order geometry

The tangent bundle TkM of order k, of a smooth Banach manifoldM consists of all equivalent classes of curves that agree up to their accelerations of order k. For a Banach manifold M and k ∈ N ∪ {0} we prove that TkM admits a vector bundle structure over M if and only if M is endowed with a linear connection. This bundle structure heavily depends on the choice of the connection. Then we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the k’th order differential Tkg : TkM −→ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. Finally, we construct the principal bundle of linear frames LkM of the vector bundle TkM over M and we prove that it is the associated bundle to TkM with respect to the identity representation of GL(Ek).