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A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-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.

Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space