Almost positive curvature on an irreducible compact rank 2 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.

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