Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them

TL;DR

This paper explores the gluing operation for diffeological vector pseudo-bundles, investigates pseudo-metrics on them, and relates these concepts to classical vector bundles, providing new insights into their structure and duality.

Contribution

It develops a detailed understanding of the gluing operation for diffeological pseudo-bundles and introduces the concept of pseudo-metrics in this context, connecting to classical bundles.

Findings

Gluing operation behavior detailed with respect to smooth maps.

Finite atlas vector bundles can be viewed as diffeological gluings.

Pseudo-metrics may not always exist, related to non-local-triviality.

Abstract

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from arXiv:1509.03023, and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a diffeological gluing, and thus deduce that its usual dual bundle is the same as its diffeological dual. We then consider the notion of a pseudo-metric, the fact that it does not always exist (which seems to be related to non-local-triviality condition), construction of an induced pseudo-metric on a pseudo-bundle obtained by gluing, and finally, the relation between the spaces of all pseudo-metrics on the factors of a gluing, and on its result. We conclude by commenting on the induced pseudo-metric on the pseudo-bundle dual to the given one.