Three new almost positively curved manifolds

TL;DR

This paper introduces three new examples of almost positively curved manifolds and analyzes the curvature properties of certain quotients, expanding the understanding of almost positive curvature in Riemannian geometry.

Contribution

The authors discover three new almost positively curved manifolds and demonstrate that a known quasi-positive curvature metric is not almost positively curved for certain cases.

Findings

Identified three new almost positively curved manifolds: $Sp(3)/Sp(1)^2$ and two circle quotients.

Proved that the quasi-positive curvature metric on $Sp(n+1)/Sp(n-1) Sp(1)$ is not almost positively curved for $n\,geq\,3$.

Abstract

A Riemannian manifold is called almost positively curved if the set of points for which all $2$-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: $Sp(3)/Sp(1)^2$, and two circle quotients of $Sp(3)/Sp(1)^2$. We also show the quasi-positively curved metric of Tapp [26]} on $Sp(n+1)/Sp(n-1) Sp(1)$ is not almost positively curved if $n\geq 3$.