An ILB- Manifold Structure on the Set of Riemannian Metrics on a   Noncompact Manifold
  In this paper, using the structures of cone and bicone fields on vector bundles, the author introduces a ILB (inverse limit of   Banach)- manifold structure on $\mathcal M$ the space of   Riemannian metrics on a noncompact manifold $M$. In the last section, it is proven that, this way, on the open submanifold $\mathcal{M}_{finite}$ of finite volume metrics, the canonical Riemannian metric is defined. 
Catalin C. Vasii