Finsler geometry and actions of the p-Schatten unitary groups

TL;DR

This paper studies the geometry of certain unitary groups and their actions on manifolds using Finsler metrics derived from p-Schatten norms, establishing existence, uniqueness, and minimality of geodesics.

Contribution

It introduces a natural Finsler metric on manifolds acted upon by p-Schatten unitary groups and proves the existence, uniqueness, and minimality of geodesics under this metric.

Findings

Existence of minimal geodesics in the manifold for given initial conditions.

Uniqueness of geodesics with specific initial data.

Completeness of the metric space with respect to the rectifiable distance.

Abstract

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $\delta(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ \delta(0)=x \hbox{and} \dot{\delta}(0)=X, $$ which remains minimal as long as $t\|z\|_p\le \pi/4$. Moreover, $\delta$ is unique with these properties. We also show that the metric space $({\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\cal O}=\{uAu^*: u\in U_p(H)\} $$ of a self-adjoint operator $A\in B(H)$.