Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature

TL;DR

This paper demonstrates that in all odd dimensions greater than or equal to five, there exist large classes of closed manifolds with infinitely many distinct positive Ricci curvature geometries, indicating a rich diversity of such metrics.

Contribution

The authors construct examples of odd-dimensional manifolds with infinitely many disconnected components in their positive Ricci curvature metric moduli space, revealing new complexity in geometric structures.

Findings

Existence of infinitely many geometries of positive Ricci curvature in odd dimensions ≥ 5

Construction of large classes of manifolds with disconnected moduli spaces

Demonstration of the richness of positive Ricci curvature geometries in odd dimensions

Abstract

In every odd dimension $n\geq 5$ we exhibit large classes of closed $n$-dimensional manifolds which admit infinitely many different geometries of positive Ricci curvature, i.e., manifolds for which their moduli space of metrics of positive Ricci curvature has infinitely many connected components.