Geodesic Flow on the Diffeomorphism Group of the circle

TL;DR

This paper investigates the geometric structure of the diffeomorphism group of the circle using right-invariant metrics, establishing properties of the exponential map and geodesics in an infinite-dimensional setting.

Contribution

It demonstrates that certain right-invariant metrics turn the diffeomorphism group into a Riemannian manifold, extending classical geometric results to infinite dimensions.

Findings

Exponential map is a smooth local diffeomorphism.

Geodesics are length-minimizing.

The group admits a Riemannian structure with these properties.

Abstract

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.