I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

/pdf/natural-operations-in-differential-geometry-dm1bq3cz0b.pdf

Natural operations in differential geometry

Thomas's Structure Bundle for Conformal, Projective and Related Structures

The Bernstein-Gelfand-Gelfand sequences extend the complexes of
homogeneous vector bundles to curved Cartan geometries

https://www.jstor.org/stable/pdfplus/3062111.pdf

Bernstein-Gelfand-Gelfand sequences

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called |k|–grading, i.e. a grading of the form g = g−k ⊕ · · · ⊕ gk, such that no simple factor of G is of type A1. Let P be the subgroup corresponding to the subalgebra p = g0 ⊕ · · · ⊕ gk. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G,P ) and to study basic properties of these geometric

/pdf/parabolic-geometries-and-canonical-cartan-connections-3pjgjkpxmm.pdf

Parabolic geometries and canonical Cartan connections

Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.

/pdf/tractor-calculi-for-parabolic-geometries-3p1ohi504k.pdf

R. J. Baston

Papers

Almost Hermitian symmetric manifolds I local twistor theory