How to project 'circular' manifolds using geodesic distances?

Recent papers have clearly shown the advantage of using the geodesic distance instead of the Euclidean one in methods performing non-linear dimensionality reduction by means of distance preservation. This new metric greatly improves the performances of existing algorithms, especially when strongly crumpled manifolds have to be unfolded. Nevertheless, neither the Euclidean nor the geodesic distance address the issue of ‘circular’ manifolds like a cylinder or a torus. Such manifolds should ideally be torn before to be unfolded. This paper describes how this can be done in practice when using the geodesic distance