The smooth Riemannian extension problem

Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain inside a geodesically complete Riemannian manifold $(M',g')$ without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.
Comment: This article supersedes arXiv:1601.05075, which contained only Part 1. New parts dealing with the Riemannian extension problem under a control of the sectional and the Ricci curvatures have been added