We obtain $C^{infty}$-continuous paths of explicit Riemannian metrics $g_t$, $0{leq}t$ < ${varepsilon}$, whose scalar curvatures $s(g_t)$ decrease, where $g_0$ is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension ${geq}3$. Some of them yield scalar curvature decrease on a ball in the Euclidean space.