  B Wilking has recently shown that one can associate a Ricci flow invariant
cone of curvature operators $C(S)$, which are nonnegative in a suitable sense,
to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For
curvature operators of a K\"ahler manifold of complex dimension $n$, one
considers $Ad_{GL(n,\C)}$ invariant subsets $S \subset {\bf gl}(n,\C)$. In this
article we show:
  (i) If $S$ is an $Ad_{SO(n,\C)}$ subset, then $C(S)$ is contained in the cone
of curvature operators with nonnegative isotropic curvature and if $S$ is an
$Ad_{GL(n,\C)}$ subset, then $C(S)$ is contained in the cone of K\"ahler
curvature operators with nonnegative orthogonal bisectional curvature.
  (ii) If $S \subset {\bf so}(n,\C)$ is a closed $Ad_{SO(n,\C)}$ invariant
subset and $C_+(S) \subset C(S)$ denotes the cone of curvature operators which
are {\it positive} in the appropriate sense then one of the two possibilities
holds: (a) The connected sum of any two Riemannian manifolds with curvature
operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$
(b) The normalized Ricci flow on any compact Riemannian manifold $M$ with
curvature operator in $C_+(S)$ converges to either to a metric of constant
positive sectional curvature or constant positive holomorphic sectional
curvature or $M$ is a rank-1 symmetric space.
