Four-dimensional Einstein manifolds with sectional curvature bounded from above

TL;DR

This paper classifies four-dimensional Einstein manifolds with positive scalar curvature under sectional curvature bounds, proving nonnegativity and rigidity results that identify these manifolds as standard spaces like spheres and projective spaces.

Contribution

It establishes new curvature bounds and rigidity theorems for four-dimensional Einstein manifolds, characterizing them as standard symmetric spaces under certain curvature conditions.

Findings

Sectional curvature bounded above implies nonnegativity.

Rigidity theorems identify manifolds as S^4, RP^4, or CP^2.

Specific curvature bounds are derived for classification.

Abstract

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative curvature and $Ric=1$. In this paper, we firstly show that if the sectional curvature satisfies $K\le M_1= \frac{\sqrt{3}}2\approx 0.866025$, then the sectional curvature will be nonnegative. Next, we prove a family of rigidity theorems of Einstein four-manifolds with nonnegative sectional curvature, and satisfies $K_{ik}+sK_{ij}\ge K_s = \frac{1 + \sqrt{2}}3 - \frac{\sqrt{4+2\sqrt{2}}}4 + \frac{2-\sqrt{2}}6 s$ for every orthonormal basis $\{e_i\}$ with $K_{ik}\ge K_{ij}$, where $s$ is any nonnegative constant. Indeed, we will show that these Einstein manifolds must be isometric either $S^4$, $RP^4$ or $CP^2$ with standard metrics. As a corollary, we give a rigidity result of Einstein four-manifolds with $Ric=1$, and the sectional curvature satisfies $K \le M_2 = \frac {2-\sqrt{2}}6 + \frac{\sqrt{4+2\sqrt{2}}}4 \approx 0.750912$.