Three new almost positively curved manifolds

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 ≥ 3.