Three new examples of compact manifolds admitting Riemannian structures of positive curvature

The purpose of this note is to announce several new results on compact homogeneous spaces that admit invariant Riemannian structures of strictly positive curvature (that is, all the sectional curvatures are bounded below by a positive constant). The main result that we announce is that the manifolds of flags in complex, quaternionic and Cayley three-space admit homogeneous Riemannian structures of strictly positive curvature. As coset spaces these manifolds are given as SU(3)/T (Ta maximal torus of SU(3)(3)/SU(2) x SU(2) x S/(2)andF 4 /Spin(8) (note that F 4 /Spin( The example F 4 /Spin(8), which we have called the manifold of flags in Cayley three-space was not on our list in We take this chance to correct our error by including the pair (F 4 , Spin(8)) in the list of Theorem 5.1 of