Abstract
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.
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Suri, A. Higher Order Tangent Bundles. Mediterr. J. Math. 14, 5 (2017). https://doi.org/10.1007/s00009-016-0812-7
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DOI: https://doi.org/10.1007/s00009-016-0812-7