On the classification of positive quaternionic Kähler 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 \( \left[ {\tfrac{m} {2}} \right] + 2 \) and the fourth Betti number b4 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 b4(M) = 1 is greater than or equal to \( \tfrac{m} {2} + 1 \) 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.