Nonnegatively curved homogeneous metrics in low dimensions

TL;DR

This paper investigates invariant Riemannian metrics on low-dimensional compact homogeneous spaces, focusing on conditions under which deformations preserve nonnegative curvature, extending known results through detailed algebraic analysis.

Contribution

It provides a comprehensive analysis of all G-invariant fibration metrics on low-dimensional homogeneous spaces, identifying algebraic criteria for maintaining nonnegative curvature during deformations.

Findings

Identifies algebraic conditions for nonnegative curvature in low-dimensional cases

Classifies invariant metrics on spaces with dim ≤ 15

Extends previous work on Cheeger deformations and fiber enlargements

Abstract

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.