Riemannian metrics on an infinite dimensional symplectic group

TL;DR

This paper explores the geometric structure of an infinite-dimensional symplectic group, analyzing Riemannian metrics, geodesics, and completeness properties within this mathematical framework.

Contribution

It introduces and compares different Riemannian metrics on the symplectic group of Hilbert-Schmidt perturbations, and studies geodesic completeness in this context.

Findings

Comparison of geodesic lengths using minimal curves of related groups

Analysis of the completeness of the symplectic group under various metrics

Characterization of geodesic curves in the infinite-dimensional setting

Abstract

The aim of this paper is the geometric study of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator. This subgroup of the symplectic group was introduced in Pierre de la Harpe's classical book of Banach-Lie groups. Throughout this paper we will endow the tangent spaces with different Riemannian metrics. We will use the minimal curves of the unitary group and the positive invertible operators to compare the length of the geodesic curves in each case. Moreover we will study the completeness of the symplectic group with the geodesic distance.