Geometric structure for the tangent bundle of direct limit manifolds

TL;DR

This paper develops a geometric framework for the tangent bundle of direct limit manifolds, equipping it with a convenient vector bundle structure using an infinite-dimensional general linear group.

Contribution

It introduces a novel geometric structure for tangent bundles of direct limit manifolds, extending finite-dimensional concepts to infinite-dimensional settings.

Findings

Established a convenient vector bundle structure for tangent bundles of direct limit manifolds.

Defined the structural group as an infinite-dimensional general linear group.

Extended finite-dimensional geometric concepts to infinite-dimensional manifolds.

Abstract

We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.