Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

TL;DR

This paper proves that the right-invariant $L^2$-metric on the Virasoro-Bott group, which relates to the KdV-equation, results in a metric space with zero geodesic distance, highlighting a geometric property of this infinite-dimensional space.

Contribution

It establishes the vanishing geodesic distance for the Virasoro-Bott group with the $L^2$-metric, connecting geometric analysis with the KdV-equation.

Findings

The $L^2$-metric on the Virasoro-Bott group has zero geodesic distance.

The result links the geometric structure to the integrability of the KdV-equation.

This contributes to understanding the geometry of infinite-dimensional groups with weak Riemannian metrics.

Abstract

The Virasoro-Bott group endowed with the right-invariant $L^2$-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.