Reduction for natural operators on projectable connections

W M Mikulski,J Tomáš

doi:10.1515/dema-2013-0165

Abstract

AbstractWe present a very simple proof of a general reduction for natural operators on torsion free projectable classical linear connections.

Highlights

A reduction for natural operators on classical linear connections on manifolds is a very old and known fact, but the known for the authors proof of it is very complicated
An F Mm,n-natural operator D : G × Qτ−proj F is a system of F Mm,ninvariant regular operators
The proof of Lemma 1 is standard and is presented in some previous papers. (We propose to use the well-known construction of normal coordinates of classical linear connections on manifolds and next to to use a simple observation that if ∇ is a projectable torsion-free classical linear connection on p : Y → M with the underlying connection ∇ on M the exponent Expy of ∇ at y is fibred over the exponent Expp(x) of ∇.)

Summary

Introduction

A reduction for natural operators on classical linear connections on manifolds is a very old and known fact, but the known for the authors proof of it (see e.g [1]) is very complicated. For any F Mm,n-object p : Y → M , where F Y (or GY ) is the set of all sections on F Y → Y (or GY → Y ) and Qτ−proj(Y ) is the set of all torsion free projectable classical linear connections on Y → M .

Results

Conclusion