Almost positive curvature on an irreducible compact rank 2 symmetric space

TL;DR

This paper constructs the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1, specifically on the Grassmannian of oriented 2-planes in R^7, using Lie groups and octonions.

Contribution

It provides the first known example of almost positive curvature on a higher-rank irreducible compact symmetric space, expanding the understanding of curvature properties in differential geometry.

Findings

Grassmannian of oriented 2-planes in R^7 admits an almost positively curved metric

Construction relies on Lie group G_2 and octonions

First such example for rank > 1 symmetric space

Abstract

A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than $1$. The construction and verification rely on the Lie group $\mathbf{G}_2$ and the octonions, so do not obviously generalize to any other Grassmannians.