A simplified Proof of the Hopf Conjecture

The use of the barycentre map between two copies of ℝⁿ , the first one with a metric without conjugate points, the second one with the canonical flat metric, allows to prove in a simplified way the fact that Riemannian tori without conjugate points are flat, as conjectured by Hopf in 1948 and proved definitively by Burago and Ivanov in 1994.