Rigidity of some Weyl manifolds with nonpositive sectional curvature

We provide a list of all locally metric Weyl connections with non-positive sectional curvatures on two types of manifolds, n-dimensional tori T-n and M-n = S-1 x Sn-1 with the standard conformal structures. For M-n we prove that it carries no other Weyl connections with nonpositive sectional curvatures, locally metric or not. In the case of T-n we prove the same in the more narrow class of integrable connections.