• geodesic distances
• infinite dimensional manifolds
• right-invariant Riemannian metrics
• sub-Riemannian controllability

It is well known that in a connected and finite dimensional Riemannian manifold, the infimum among the lengths of all continuous and piecewise smooth curves connecting two points yields a distance, called geodesic distance. The objective of this thesis is to understand whether the analogous procedure leads to a distance also when the model space of the manifold is infinite-dimensional. Examples of vanishing geodesic distances for Fréchet manifolds were known. In this thesis we give a new result, presenting a simple example of vanishing geodesic distance in a separable Hilbert space. We also study the case of right-invariant Riemannian metrics on Lie groups, presenting a recent result by Bauer, Harms and Preston, which characterizes the vanishing of the associated geodesic distance in terms of the action of the group on a set. Finally, we present a recent generalization of the Chow-Rashevskii theorem to horizontal vector bundles on Hilbert manifolds, proved by Arguillére.