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

Jean-Pierre Magnot

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.

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

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.
Paper Structure (22 sections, 27 theorems, 50 equations)

This paper contains 22 sections, 27 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on diffeologies and vector pseudo-bundles
  3. Basics of Diffeology
  4. Frölicher spaces
  5. Quotient and subsets
  6. Vector pseudo-bundles
  7. Tangent and cotangent spaces
  8. On examples of interest
  9. Technical constructions: $G-$diffeology and isomorphisms of vector pseudo-bundles
  10. The $G-$diffeology and its tangent space
  11. Isomorphisms of vector pseudo-bundles
  12. On the group of automorphisms and on the frame bundle of a vector pseudo-bundle
  13. On the group of automorphisms
  14. Frame bundles for diffeological vector bundles
  15. Connection 1-forms versus covariant derivatives
  16. ...and 7 more sections

Key Result

Proposition 2.3

SouIgdiff Let $(X',\mathcal{P})$ be a diffeological space, and let $X$ be a set. Let $f:X\rightarrow X'$ be a map. We define $f^*(\mathcal{P})$ the pull-back diffeology as

Theorems & Definitions (74)

  • Definition 2.1: Diffeology
  • Definition 2.2: Diffeologically Smooth Map
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6: Product Diffeology
  • Definition 2.7: Subset Diffeology
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • ...and 64 more