Almost conformally Einstein manifolds and obstructions

TL;DR

This paper develops a systematic method to identify obstructions to conformally Einstein metrics using tractor calculus, and introduces a generalized Einstein equation by relaxing certain conditions.

Contribution

It provides a new framework for constructing obstructions to conformally Einstein structures through tractor bundles and generalizes Einstein equations.

Findings

Systematic approach to obstructions using tractor calculus

Introduction of a generalized Einstein equation

Identification of conditions for conformally Einstein metrics

Abstract

A Riemannian or pseudo-Riemannian (or conformal) structure is conformally Einstein if and only if there is a suitably generic parallel section of a certain vector bundle -- the so-called standard conformal tractor bundle. We show that this characterisation leads to a systematic approach to constructing obstructions to conformally Einstein metrics. Relaxing the requirement that the parallel tractor field be generic gives a natural generalisation of the Einstein equations.