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

Abstract

In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^{r}M$, $r\in \mathbb{N}\cup \{\infty \}$. First we endow $T^{r}M$ with a canonical atlas using that of $M$. Then the concepts of vertical and complete lifts for functions and vector fields on $T^{r}M$ 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 Fréchet 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.

References in corpus (1)

• k-Parameter geodesic variations

Cited by in corpus (1)

• Basic curvature and the Atiyah cocycle in gauge theory