Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them
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.
Keywords
Cite
@article{arxiv.1601.00170,
title = {Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them},
author = {Ekaterina Pervova},
journal= {arXiv preprint arXiv:1601.00170},
year = {2016}
}
Comments
31 pages; title changed, an omission in the proof of Theorem 2.3 corrected, an erroneous claim (that was not used anywhere in the paper) taken out of the statement of Lemma 4.1, Section 3 shortened for expository purposes, various minor improvements throughout