Metrics in the space of curves

TL;DR

This paper explores geometries on the manifold of curves, defining metrics like Riemannian and Finsler, studying minimal geodesics, and proposing a conformal version of the $H^0$ metric for curve deformations.

Contribution

It introduces a geometric framework for the manifold of curves with new metrics and analyzes minimal geodesics under constraints, including a conformal $H^0$ metric.

Findings

Analysis of the $H^0$ metric and its properties

Existence of minimal geodesics under constraints

Proposal of a conformal $H^0$ metric

Abstract

In this paper we study geometries on the manifold of curves. We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to \real^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to\real^n$ of $c$. We discuss Riemannian and Finsler metrics $F(c,h)$ on this manifold $M$, and in particular the case of the geometric $H^0$ metric $F(c,h)=\int |h|^2ds$ of normal deformations $h$ of $c$; we study the existence of minimal geodesics of $H^0$ under constraints; we moreover propose a conformal version of the $H^0$ metric.