The curvature of convex sum of metrics and applications

TL;DR

This paper derives explicit formulas for the curvature of convex combinations of Riemannian metrics and investigates conditions under which such deformations can increase the average curvature along flat tori.

Contribution

It provides necessary and sufficient conditions for convex metric deformations to increase average curvature, with applications to classical metric changes.

Findings

First-order increase in curvature is prohibited.

Necessary and sufficient conditions for higher-order positive average curvature variation.

Applications to conformal, warping, and Cheeger deformations.

Abstract

In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \(g_t = (1-t)g_0 + t g_1\). We study whether such a deformation can increase the \emph{average} of the Riemann curvature component \(R_t(X,Y,Y,X)\) along an immersed, totally geodesic flat torus. Because a first-order increase is prohibited, we obtain necessary and sufficient conditions for \(g_t\) to have a positive average variation of order \(r \geq 2\). These conditions are applied to paths joining \(g_0\) to classical metric deformations, including conformal changes, vertical warpings, and Cheeger deformations.