Homogeneous Metrics with nonnegative curvature

TL;DR

This paper investigates conditions under which enlarging subgroups within compact Lie groups preserves nonnegative curvature, leading to new examples of homogeneous spaces with such metrics.

Contribution

It establishes criteria for subgroup enlargements that maintain nonnegative curvature and provides new examples of nonnegatively curved homogeneous metrics.

Findings

Enlarging subgroups in symmetric pairs preserves nonnegative curvature.

Identifies conditions for maintaining nonnegative curvature during subgroup enlargement.

Provides numerous new examples of homogeneous spaces with nonnegative curvature.

Abstract

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.