The curvature of convex sum of metrics and applications

Abstract

In this paper we explore the curvature of convex sums of Riemannian metrics g(t) = (1 − t)g0 + tg1, 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 ≥ 2) along the torus and apply them for paths joining g0 to known classical deformations.