Higher Order Tangent Bundles
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- 作者：Ali Suri
- 关键词：Banach manifold ; linear connection ; connection map ; higher order tangent bundle ; Fréchet manifold ; lifting of Riemannian metrics ; Lagrangians
- 刊名：Mediterranean Journal of Mathematics
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：14
- 期：1
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics, general;
- 出版者：Springer International Publishing
- ISSN：1660-5454
- 卷排序：14
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.