Definite signature conformal holonomy: a complete classification

TL;DR

This paper provides a complete classification of conformal Tractor holonomy, linking it to the underlying manifold's geometry, especially in Einstein and Ricci-flat cases, using metric cone constructions and decomposition theorems.

Contribution

It offers a comprehensive classification of conformal Tractor holonomy, connecting it to geometric structures and extending previous partial results.

Findings

Holonomy classification for conformal Tractor connections.

Relation between Tractor holonomy and Einstein geometry.

Decomposition theorem for Ricci-flat cases.

Abstract

This paper aims to classify the holonomy of the conformal Tractor connection, and relate these holonomies to the geometry of the underlying manifold. The conformally Einstein case is dealt with through the construction of metric cones, whose Riemmanian holonomy is the same as the Tractor holonomy of the underlying manifold. Direct calculations in the Ricci-flat case and an important decomposition theorem complete the classification for definitive signature.