Mixed sectional-Ricci curvature obstructions on tori

TL;DR

This paper proves new geometric obstructions on tori, showing that certain curvature bounds cannot be simultaneously satisfied, thus advancing understanding of Riemannian metrics with mixed curvature conditions.

Contribution

It establishes explicit obstructions to the coexistence of bounded sectional and Ricci curvatures on tori, extending previous results on curvature constraints.

Findings

Negative Ricci curvature on tori implies positive sectional curvature in some directions.

Sectional curvature cannot be arbitrarily small if Ricci curvature is bounded away from zero.

Constants depend only on the dimension of the torus.

Abstract

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.