Florent Tallerie [1]
A famous theorem of Nash, improved by Kuiper, implies that every Riemannian surface admits a C1 isometric embedding in the three dimensional Euclidean space. A similar result due to Burago and Zalgaller states that every polyhedral surface, that is a surface obtained from gluing Euclidean polygons, admits a piecewise linear (in short PL) isometric embedding in the three dimensional Euclidean space. In particular, any flat torus, that is a quotient of the Euclidean plane by a rank two lattice, admits such a PL isometric embedding. Nevertheless, the proof of Burago and Zalgaller is partially not constructive, relying on the Nash-Kuiper process. We will see how to make it effective in the case of flat tori.