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Lie derivatives of sections of natural vector bundles

Peter W. Michor

Abstract

Time derivatives of pullbacks and push forwards along smooth curves of diffeomorphism of sections of natural vector bundles are computed in terms of Lie derivatives along adapted non-autonomous vector fields by extending a key lemma in [Markus Mauhart, Peter W. Michor: Commutators of flows and fields. Arch. Math. (Brno) 28 (1992), 228-236. arXiv:math/9204221]. There is also the analogous result about the first non-vanishing derivative of higher order.

Lie derivatives of sections of natural vector bundles

Abstract

Time derivatives of pullbacks and push forwards along smooth curves of diffeomorphism of sections of natural vector bundles are computed in terms of Lie derivatives along adapted non-autonomous vector fields by extending a key lemma in [Markus Mauhart, Peter W. Michor: Commutators of flows and fields. Arch. Math. (Brno) 28 (1992), 228-236. arXiv:math/9204221]. There is also the analogous result about the first non-vanishing derivative of higher order.

Paper Structure

This paper contains 6 sections, 5 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Background from MauhartMichor92
  3. Curves of local diffeomorphisms
  4. Natural vector bundles
  5. Pullback of sections
  6. The results

Key Result

Corollary 1

Let $\varphi_t$ be a smooth curve of local diffeomorphisms. Then we get two time dependent vector fields Then for any natural vector bundle functor $F$ and for any section $s\in \Gamma(F(M))$ we have the first non-vanishing derivative

Theorems & Definitions (8)

  • Corollary
  • Lemma 2.2
  • proof
  • Lemma 2.5
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof