Metrics of Positive Ricci Curvature on Simply-Connected Manifolds of Dimension $6k$

TL;DR

This paper introduces a new method to construct metrics of positive Ricci curvature on certain 6k-dimensional manifolds, expanding the known examples and understanding of such geometries.

Contribution

The paper develops a novel graph-based description of 6k-dimensional manifolds and uses it to construct new positive Ricci curvature metrics, filling gaps in existing knowledge.

Findings

Constructed many new examples of 6k-dimensional manifolds with positive Ricci curvature.

Provided a new description of manifolds via labeled bipartite graphs.

Extended the class of known manifolds with positive Ricci curvature.

Abstract

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article we introduce a new description of certain $6k$-dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way we obtain many new examples, both spin and non-spin, of $6k$-dimensional manifolds with a metric of positive Ricci curvature.