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Notes on Tractor Calculus

Jan Slovák, Radek Suchánek

TL;DR

This survey provides an elementary yet comprehensive introduction to tractor calculus within conformal and parabolic (Cartan) geometries, emphasizing the construction of tractor bundles from a Cartan connection and the role of the Maurer–Cartan form in modeling the conformal sphere. It develops the Thomas tractor framework, the Weyl and $\mathsf P$-corrected connections, and the prolongation techniques that convert conformally invariant PDEs into parallel-tractor conditions, notably the conformal to Einstein problem. The text then situates tractors in the broader parabolic geometry setting, detailing $|1|$-graded structures, natural/adjoint tractors, the fundamental derivative, and the BGG machinery, including normalization via harmonic curvature and Kostant cohomology. Concrete examples (conformal, projective, almost Grassmannian) illustrate how these tools yield invariant differential operators, such as the Yamabe operator, and provide a path to systematic, manifestly invariant calculus on curved geometries. Altogether, the work outlines a unifying framework for deriving higher-order, invariant operators on Cartan geometries and clarifies the algebraic underpinnings behind normalization, deformations, and the BGG sequences.

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.

Notes on Tractor Calculus

TL;DR

This survey provides an elementary yet comprehensive introduction to tractor calculus within conformal and parabolic (Cartan) geometries, emphasizing the construction of tractor bundles from a Cartan connection and the role of the Maurer–Cartan form in modeling the conformal sphere. It develops the Thomas tractor framework, the Weyl and -corrected connections, and the prolongation techniques that convert conformally invariant PDEs into parallel-tractor conditions, notably the conformal to Einstein problem. The text then situates tractors in the broader parabolic geometry setting, detailing -graded structures, natural/adjoint tractors, the fundamental derivative, and the BGG machinery, including normalization via harmonic curvature and Kostant cohomology. Concrete examples (conformal, projective, almost Grassmannian) illustrate how these tools yield invariant differential operators, such as the Yamabe operator, and provide a path to systematic, manifestly invariant calculus on curved geometries. Altogether, the work outlines a unifying framework for deriving higher-order, invariant operators on Cartan geometries and clarifies the algebraic underpinnings behind normalization, deformations, and the BGG sequences.

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.

Paper Structure

This paper contains 27 sections, 12 theorems, 102 equations.

Table of Contents

  1. Tracy Thomas' conformal tractors
  2. Riemannian sphere
  3. Conformal Riemannian sphere
  4. Towards tractors
  5. Tracy Thomas' tractors $\mathcal{T}$
  6. Conformal to Einstein and the tractor connection
  7. The Einstein scales
  8. Conformal invariance
  9. Parabolic geometries
  10. $|1|$-graded parabolic geometries
  11. Natural bundles and Weyl connections
  12. Higher order derivatives
  13. A few examples
  14. Elements of tractor calculus
  15. Natural bundles and tractors
  16. ...and 12 more sections

Key Result

theorem thmcountertheorem

Let $\omega_G$ be the Maurer-Cartan form of a Lie group $G$ with the Lie algebra $\mathfrak{g}$, $M$ a smooth manifold endowed with a $1$-form $\omega \in \Omega^1(M, \mathfrak{g})$. Then for each $x \in M$ there is a neighborhood $U \ni x$ and $f \colon U \to G$ such that $f^{*} \omega_G=\omega$, i If $M$ is connected and $f_1, f_2 \colon M \to G$ with $f_1^{*} \omega_G = f_2^{*} \omega_G$ on $M$

Theorems & Definitions (22)

  • definition thmcounterdefinition
  • theorem thmcountertheorem: Fundamental theorem of calculus
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • definition thmcounterdefinition
  • ...and 12 more