Closed $G_2$-Structures with Negative Ricci Curvature
Differential Geometry
2025-10-07 v3
Abstract
We study existence problems for closed -structures with negative Ricci curvature, and we prove the -Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed -structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed -structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed -structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.
Cite
@article{arxiv.2310.19553,
title = {Closed $G_2$-Structures with Negative Ricci Curvature},
author = {Alec Payne},
journal= {arXiv preprint arXiv:2310.19553},
year = {2025}
}
Comments
Minor typos fixed, published in Bull. Lond. Math. Soc