Estimates of Gromov's box distance

TL;DR

This paper estimates Gromov's box distance between spheres, complex projective spaces, and special orthogonal groups, providing explicit bounds and completing a problem posed in Gromov's foundational work.

Contribution

It introduces new estimates for Gromov's box distance on key geometric spaces, solving an open exercise from Gromov's book.

Findings

Estimated $oxdistance(S^n,S^m)$ and $oxdistance(C P^n,C P^m)$

Provided lower bounds for $oxdistance(SO(n), SO(m))$

Solved an open problem from Gromov's Green book

Abstract

In 1999, M. Gromov introduced the box distance function $\sikaku$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding (cf. \cite[Lemma 5.10]{Colding}), we estimate $\sikaku(\mathbb{S}^n,\mathbb{S}^m)$ and $\sikaku (\mathbb{C}P^n, \mathbb{C}P^m)$, where $\mathbb{S}^n$ is the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ and $\mathbb{C}P^n$ is the $n$-dimensional complex projective space equipped with the Fubini-Study metric. In paticular, we give the complete answer to an Exercise of Gromov's Green book (cf. \cite[Section $3{1/2}.18$]{gromov}). We also estimate $\sikaku \big(SO(n), SO(m)\big)$ from below, where SO(n) is the special orthogonal group.