Compact 4-Dimensional Spin Gradient $m$-quasi-Einstein Manifolds Satisfy the Hitchin-Thorpe Inequality when $m\ge 1$

TL;DR

This paper proves that certain 4-dimensional spin gradient m-quasi-Einstein manifolds satisfy the Hitchin-Thorpe inequality and classifies their universal covers as either S^4 or connected sums of S^2×S^2.

Contribution

It establishes the Hitchin-Thorpe inequality for 4D spin gradient m-quasi-Einstein manifolds with m≥1 and classifies their universal cover topology.

Findings

Manifolds satisfy the Hitchin-Thorpe inequality.

Universal cover is either S^4 or a connected sum of S^2×S^2.

Results hold for manifolds with nontrivial potential functions.

Abstract

We prove that a compact, connected, and oriented 4-dimensional gradient $m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that the homeomorphism-type of the universal cover of such a manifold is either $S^4$ or a connected sum of some number of $S^2\times S^2$ when the potential function is nontrivial.