Projective Geometry II: Cones and Complete Classifications

TL;DR

This paper classifies irreducible holonomy algebras of the projective Tractor connection by constructing Ricci-flat cones, enabling the creation of manifolds with all possible holonomy algebras.

Contribution

It introduces a novel approach using projective cones to classify and construct manifolds with specific holonomy algebras, extending previous work.

Findings

Constructed Ricci-flat cones corresponding to Tractor holonomies

Achieved classification of irreducible holonomy algebras

Enabled explicit manifold constructions for each holonomy

Abstract

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.