Projective Geometry I: Principles and Properties

Abstract

The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations. Preserved subbundles of the Tractor bundle generate foliations with Ricci-flat leaves. Contact- and Einstein-structures arise from other reductions of the Tractor holonomy, as do U(1) and $Sp(1,\mathbb{H})$ bundles over a manifold of smaller dimension.

27 pages, paper 1 of 2, slight corrections and modifications, as well as a changed division with paper 2

References in corpus (5)

• Ricci-corrected derivatives and invariant differential operators
• Contact Projective Structures
• Normal Conformal Killing Forms
• Definite signature conformal holonomy: a complete classification
• Ambient connections realising conformal Tractor holonomy