On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives

TL;DR

This paper explores the extension of classical differential geometry concepts like automorphisms and connections to the broader setting of diffeological vector pseudo-bundles, including infinite-dimensional cases, highlighting key distinctions and applications.

Contribution

It introduces a framework for differential geometric tools on diffeological vector pseudo-bundles, emphasizing differences from classical theory and extending to infinite-dimensional bundles.

Findings

Identifies non-isomorphism between connection 1-forms and covariant derivatives in this setting.

Extends geometric tools to infinite-dimensional vector bundles.

Provides examples with singularities and applications to infinite-dimensional bundles.

Abstract

We consider here the category of diffeological vector pseudo-bundles, and study a possible extension of classical differential geometric tools on finite dimensional vector bundles, namely, the group of automorphisms, the frame bundle, the space of connection 1-forms and the space of covariant derivatives. Substential distinctions are highlighted in this generalized framework, among which the non-isomorphism between connection 1-forms and covariant derivatives. Applications not only include finite dimensional examples with singularities, but also infinite dimensional vector bundles.