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

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.

33 pages; final version

References in corpus (7)

• Smooth metric measure spaces with non-negative curvature
• Invariant Prolongation of BGG-Operators in Conformal Geometry
• Uniqueness of warped product Einstein metrics and applications
• Almost Einstein and Poincare-Einstein manifolds in Riemannian signature
• On the classification of warped product Einstein metrics
• The geometry of conformally Einstein metrics with degenerate Weyl tensor
• Warped product Einstein metrics over spaces with constant scalar curvature

Cited by in corpus (7)

• The weighted ${\sigma }_k$-curvature of a smooth metric measure space
• A notion of the weighted ${\sigma }_k$-curvature for manifolds with density
• A generalization of the Cai--Galloway splitting theorem to smooth metric measure spaces
• Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian
• The Weighted Ambient Metric
• Deformation of the Weighted Scalar Curvature
• Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev inequalities