2001/11/20 by Andreas Čap, A. Rod Gover · 8 citations
Mathematics · #Advanced Algebra and Geometry #Geometric and Algebraic Topology #Advanced Operator Algebra Research #Algorithm #Type (biology) #Artificial intelligence #Mathematics #Computer science #Geology
paper · pdf · doi:10.1090/s0002-9947-01-02909-9
openalex publication_date 2001/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
Parabolic geometries may be considered as curved analogues of the homogeneous spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper P"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G/P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a semisimple Lie group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P subset-of upper G"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">P⊂ G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.