Smooth metric measure spaces, quasi-Einstein metrics, and tractors

TL;DR

This paper applies tractor calculus from conformal geometry to smooth metric measure spaces, establishing a correspondence with quasi-Einstein metrics and deriving bounds on their space dimension.

Contribution

It introduces a tractor formalism approach to study quasi-Einstein metrics, offering new insights and bounds not previously available.

Findings

Establishes a correspondence between quasi-Einstein metrics and tractor bundle sections

Provides a sharp upper bound on the dimension of the space of quasi-Einstein metrics

Offers a new perspective on recent results by He, Petersen, and Wylie

Abstract

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.