Rigidity of the stable norm on tori

Given a closed, orientable Riemannian manifold, we study the stable norm on the real homology groups. In particular, for $n\geq 2$ we prove that a Riemannian $n$-torus, which has the same stable norms as a flat $n$-torus on the first and $n-1$ homology groups, is in fact isometric to the flat torus.