Beginning with this chapter, we work on Riemannian manifolds that are either compact or with cylindrical ends. This chapter develops the theory of layer potentials for a generalized, smooth-coefficient Brinkman operator on a closed Riemannian manifold. The Brinkman operator is a compact perturbation of the Stokes operator, because it is a perturbation of the Stokes operator by suitable lower order operators. Our approach is to first use the theory of pseudodifferential and Fredholm operators to construct the fundamental solution of the generalized Brinkman operator. We then use this fundamental solution to construct the layer potentials associated to the generalized Brinkman operator on a Lipschitz domain in a closed manifold and to prove various mapping properties of these potentials.