An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures

Introduction. The purpose of this note is to show that if T 1 is a closed, connected one-dimensional subgroup of SU(3) that has no nonzero fixed points, then SUtyjT 1 admits an SC/^-invariant Riemannian structure of strictly positive curvature. This result implies the statement of the title since there are an infinite number of distinct homotopy types among the spaces SUtyjT 1 with T 1 as above (see Lemma 3.3).