Positive quaternionic Kaehler manifolds and symmetry rank

Abstract

A quaternionic K\"ahler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic K\"ahler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic K\"ahler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic K\"ahler manifold with symmetry rank at least (m-2) must be either isometric to \Bbb HP^m or Gr_2(\Bbb C^{m+2}), if m\ge 10.