Complete lift of vector fields and sprays to $T^\infty M$

TL;DR

This paper develops a geometric framework for iterated tangent bundles of Banach manifolds, defining lifts of functions and vector fields, and extends these concepts to infinite-dimensional manifolds, including applications to the manifold of closed curves.

Contribution

It introduces a canonical atlas for iterated tangent bundles of Banach manifolds and extends the concepts of lifts and sprays to infinite-dimensional settings.

Findings

Defined vertical and complete lifts for functions and vector fields on $T^rM$.

Established a generalized Fréchet manifold structure on $T^ ext{infty} M$.

Demonstrated lifting of vector fields and sprays to $T^ extinfty M$ and analyzed ODEs in this context.

Abstract

In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in {\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of $M$. Then the concepts of vertical and complete lifts for functions and vector fields on $T^rM$ are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply $T^\infty M$ with a generalized Fr\'{e}chet manifold structure and we will show that any vector field or (semi)spray on $M$, can be lifted to a vector field or (semi)spray on $T^\infty M$. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on $T^\infty M$ including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite dimensional case of manifold of closed curves.