文摘
The tangent bundle \(T^kM\) of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k. For a Banach manifold M and a natural number k, first we determine a smooth manifold structure on \(T^kM\) which also offers a fiber bundle structure for \((\pi _k,T^kM,M)\). Then we introduce a particular lift of linear connections on M to geometrize \(T^kM\) as a vector bundle over M. More precisely based on this lifted nonlinear connection we prove that \(T^kM\) admits a vector bundle structure over M if and only if M is endowed with a linear connection. As a consequence, applying this vector bundle structure we lift Riemannian metrics and Lagrangians from M to \(T^kM\). In addition, using the projective limit techniques, we declare a generalized Fréchet vector bundle structure for \(T^\infty M\) over M.
