Einstein Constants and Smooth Topology

TL;DR

This paper reviews progress on Einstein metrics with opposite Ricci signs on high-dimensional smooth manifolds and presents new related results to encourage further research in the field.

Contribution

It provides a comprehensive review and introduces new results related to Einstein metrics with opposite Ricci signs on smooth manifolds.

Findings

High-dimensional manifolds admit Einstein metrics with Ricci curvatures of opposite signs

New results related to Einstein metrics and smooth topology are established

The paper aims to stimulate further research in Einstein geometry

Abstract

It was first shown in (Catanese-LeBrun 1997) that certain high-dimensional smooth closed manifolds admit pairs of Einstein metrics with Ricci curvatures of opposite sign. After reviewing subsequent progress that has been made on this topic, we then prove various related results, with the ultimate goal of stimulating further research on associated questions.