Higher order tangent bundles

Abstract

The tangent bundle $T^{k}M$ 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 $T^{k}M$ which also offers a fiber bundle structure for $({\pi }_k,T^{k}M,M)$. Then we introduce a particular lift of linear connections on $M$ to geometrize $T^{k}M$ as a vector bundle over $M$. More precisely based on this lifted nonlinear connection we prove that $T^{k}M$ 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 $T^{k}M$. Also, using the projective limit techniques, we declare a generalized Fréchet vector bundle structure for $T^{\infty }M$ over $M$.

References in corpus (1)

• Isomorphism classes for higher order tangent bundles

Cited by in corpus (4)

• Reductions of Topologically Massive Gravity I: Hamiltonian Analysis of The Second Order Degenerate Lagrangians
• Reductions of Topologically Massive Gravity II: First Order Realizations of Second Order Lagrangians
• Second order time dependent tangent bundles and their applications
• Isomorphism classes for higher order tangent bundles