  Let $M$ be a graph manifold containing a single JSJ torus $T$ and whose JSJ
blocks are of the form $\Sigma \times S^1$, where $\Sigma$ is a compact
orientable surface with boundary. We show that if $M$ does not admit a
Riemannian metric of everywhere nonpositive sectional curvature, then there is
an essential curve on $T$ such that any finite-dimensional linear
representation of $\pi_1(M)$ maps an element representing that curve to a
matrix all of whose eigenvalues are roots of $1$. In particular, this shows
that $\pi_1(M)$ does not admit a faithful finite-dimensional unitary
representation, and gives a new proof that $\pi_1(M)$ is not linear over any
field of positive characteristic.
