Obstructions to distinguished Riemannian metrics via Lorentzian geometry

TL;DR

This paper develops a Lorentzian geometric framework to identify obstructions to special Riemannian metrics, providing new local and global criteria for Ricci-flatness, local symmetry, and constant curvature in various manifolds.

Contribution

It introduces a duality-based approach linking Lorentzian and Riemannian metrics to find obstructions to curvature conditions, including new local and global criteria.

Findings

Necessary local conditions for Ricci-flat and symmetric Riemannian metrics

Obstructions to Riemannian metrics with certain curvature properties

Non-existence of certain compact Riemannian manifolds with non-constant curvature

Abstract

We approach the problem of finding obstructions to curvature distinguished Riemannian metrics by considering Lorentzian metrics to which they are dual in a suitable sense. Obstructions to the latter then yield obstructions to the former. This framework applies both locally and globally, including to compact manifolds, and is sensitive to various aspects of curvature. Here we apply it in two different ways. First, by embedding a Riemannian manifold into a Lorentzian one and utilizing Penrose's "plane wave limit," we find necessary local conditions, in terms of the Hessian of just one function, for large classes of Riemannian metrics to contain within them those that have parallel Ricci tensor, or are Ricci-flat, or are locally symmetric. Second, by considering Riemannian metrics dual to constant curvature Lorentzian metrics via a type of Wick rotation, we are able to rule out the existence of a family of compact Riemannian manifolds (in all dimensions) that deviate from constant curvature in a precise sense.