  In this paper we construct Ricci-positive metrics on the connected sum of
products of arbitrarily many spheres provided the dimensions of all but one
sphere in each summand are at least 3. There are two new technical theorems
required to extend previous results on sums of products of two spheres. The
first theorem is a gluing construction for Ricci-positive manifolds with
corners generalizing a gluing construction of Perelman for Ricci-positive
manifolds with boundaries. Our construction gives a sufficient condition for
gluing together two Ricci-positive manifolds with corners along isometric faces
so that the resulting smooth manifold with boundary will be Ricci-positive and
have convex boundary. The second theorem claims that one can deform the
boundary of a Ricci-positive Riemannian manifold with convex boundary along a
Ricci-positive isotopy while preserving Ricci-positivity and boundary
convexity.
