Diameters of Homogeneous Spaces

TL;DR

This paper establishes a uniform lower bound on the diameter of homogeneous spaces formed by quotienting a compact Lie group by its proper closed subgroups, using a novel bi-invariant metric derived from the group's action.

Contribution

It introduces a new operator norm-based metric on compact Lie groups and proves a universal lower bound on the diameters of their homogeneous quotients.

Findings

Existence of a universal constant   for diameter bounds

The metric is bi-invariant and derived from the adjoint action

Lower bound applies to all proper closed subgroups

Abstract

Let G be a compact connected Lie group with trivial center. Using the action of G on its Lie algebra, we define an operator norm | |_{G} which induces a bi-invariant metric d_G(x,y)=|Ad(yx^{-1})|_{G} on G. We prove the existence of a constant \beta \approx .12 (independent of G) such that for any closed subgroup H \subsetneq G, the diameter of the quotient G/H (in the induced metric) is \geq \beta.