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.