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2021 | OriginalPaper | Chapter

2. Notes on Tractor Calculi

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Authors: Jan Slovák, Radek Suchánek

Publisher: Springer International Publishing

Published in: Differential Geometry, Differential Equations, and Mathematical Physics

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Abstract

These notes present elementary introduction to tractors based on classical examples, together with glimpses towards modern invariant differential calculus related to vast class of Cartan geometries, the so-called parabolic geometries.

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Here u ⋅ h is multiplication in G and h ⁻¹ ⋅ v is the left-action of H or G on \(\mathbb {V}\) given by the chosen representation.

We know that μ _a must be of weight 1 because covariant differentiation does not alter weights and σ is already of weight 1.

Recall that \(\pi \colon \mathcal {A}\mathcal {M} \to TM \) is the projection from sequence ( 2.38).

We iterate the first jet prolongation. Considering the first jets of sections of a bundle \(\mathcal W\), the jets in a fiber of \(J^1(J^1\mathcal W)\) look in coordinates as 4-tuples \((y^p,y^p_i,Y^p_j,Y^p_{ij})\) were \(Y^p_{ij}\) do not need to be symmetric. These are the non-holonomic 2-jets. The semi-holonomic ones remove part of the redundancy by requesting that the two natural projections to 1-jets coincide, i.e., \(y^p_i=Y^p_i\). This construction extends to all orders and the semi-holonomic jets look in coordinates nearly as the holonomic ones, just losing the symmetry of the derivatives. See, e.g., [ 17] for detailed exposition.

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Title

Notes on Tractor Calculi

Book

Differential Geometry, Differential Equations, and Mathematical Physics

Print ISBN: 978-3-030-63252-6

Electronic ISBN: 978-3-030-63253-3

https://doi.org/10.1007/978-3-030-63253-3

DOI

https://doi.org/10.1007/978-3-030-63253-3_2

Authors:

Jan Slovák
Radek Suchánek

Publisher

Springer International Publishing

Sequence number

Chapter number

Chapter 2

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