  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.
