On the normal functor in the category of smooth vector bundles

TL;DR

This paper investigates the properties of the normal functor in smooth vector bundles, revealing symmetry phenomena through the development of pullback and quotient theories for double vector bundles.

Contribution

It introduces a new theory of pullback and quotient for double vector bundles, explaining the naturality and symmetry of the normal functor in this context.

Findings

Symmetry phenomena occur after iterating the normal functor twice.

Pullback and quotient operations are key to understanding the normal functor's naturality.

Universal behavior and compatibility of operations explain the observed symmetry.

Abstract

This article is dedicated to the study of the normal functor in the category of smooth real vector bundles. Particularly, we focus on a symmetry phenomena which occurs after iterating two times the normal functor on a commutative square of smooth immersions. To do so, a theory of pullback and quotient is developed for double vector bundles but also for some classes of morphisms. These two operations turn out to be the key ingredients in order to study the naturality of the normal functor. The expected symmetry is then obtained thanks to the universal behavior and the mutual compatibility of these operations.