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In this paper we construct Ricci-positive metrics on the connected sum ofproducts of arbitrarily many spheres provided the dimensions of all but onesphere in each summand are at least 3. There are two new technical theoremsrequired to extend previous results on sums of products of two spheres. Thefirst theorem is a gluing construction for Ricci-positive manifolds withcorners generalizing a gluing construction of Perelman for Ricci-positivemanifolds with boundaries. Our construction gives a sufficient condition forgluing together two Ricci-positive manifolds with corners along isometric facesso that the resulting smooth manifold with boundary will be Ricci-positive andhave convex boundary. The second theorem claims that one can deform theboundary of a Ricci-positive Riemannian manifold with convex boundary along aRicci-positive isotopy while preserving Ricci-positivity and boundaryconvexity.

Ricci-positive metrics on connected sums of products with arbitrarily many spheres