  We establish new obstruction results to the existence of Riemannian metrics
on tori satisfying mixed bounds on both their sectional and Ricci curvatures.
More precisely, from Lohkamp's theorem, every torus of dimension at least three
admits Riemannian metrics with negative Ricci curvature. We show that the
sectional curvature of these metrics cannot be bounded from above by an
arbitrarily small positive constant. In particular, if the Ricci curvature of a
Riemannian torus is negative, bounded away from zero, then there exist some
planar directions in this torus where the sectional curvature is positive,
bounded away from zero. All constants are explicit and depend only on the
dimension of the torus.
