  In this paper we explore the curvature of convex sums of Riemannian metrics
$g(t) = (1-t)g_0 + tg_1$, and study whether such a deformation can increase the
curvature of an immersed totally geodesic flat tori. We obtain necessary and
sufficient conditions for $g(t)$ to have average positive $r$-order variation
($r\geq 2)$ along the torus and apply them for paths joining $g_0$ to known
classical deformations.
