Rigidity of some Weyl manifolds with nonpositive sectional curvature

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