On the Classification of Positive Quaternionic Kahler Manifolds with b(4)=1

Abstract

Let M be a positive quaternionic Kahler manifold of dimension 4m. We already showed that if the symmetry rank is greater than or equal to [m/2] + 2 and the fourth Betti number b(4) is equal to one, then M is isometric to HPm. The goal of this paper is to report that we can improve the lower bound of the symmetry rank by one for higher even-dimensional positive quaternionic Kahler manifolds. Namely, it is shown in this paper that if the symmetry rank of M with b(4)(M) = 1 is greater than or equal to m/2 + 1 for m >= 10, then M is isometric to HPm. One of the main strategies of this paper is to apply a more delicate argument of Frankel type to positive quaternionic Kahler manifolds with certain symmetry rank.