We find a $C^{infty}$-continuous path of Riemannian metrics $g_t$ on $mathbb{R}^k$, $k{geq}3$, for $0{leq}t{leq}{varepsilon}$ for some number ${varepsilon}$ > 0 with the following property: $g_0$ is the Euclidean metric on $mathbb{R}^k$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.