In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds. The methods, results and some notation of [13], hereafter referred to as GPC, and [12] will be used. These two papers and [14] can be considered as a starting point for this one. The main theorems in these papers are special cases of the theorems here. Among the most important theorems in this paper are 1. 1 and 6. 1. Some conversations with A. Haefliger were helpful in the preparation of parts of this paper. Everything will be considered from the differentable, equivalently C*, point of view; manifolds, imbeddings, and isotopes will be C*.