Geometric graph manifolds with non-negative scalar curvature

We classify n-dimensional geometric graph manifolds with non-negative scalar curvature by first showing that if n>3, the universal cover splits off a codimension 3-Euclidean factor. We then proceed with the classification of the 3-dimensional case, where the condition is equivalent to the eigenvalues of the Ricci tensor being (lambda,lambda,0) with lambda > 0. In this case we prove that such a manifold is either a lens space or a prism manifold with a very rigid metric. This allows us to also classify the moduli space of such metrics: it has infinitely many connected components for lens spaces, while it is connected for prism manifolds.