# Fractional Sobolev metrics on spaces of immersed curves

## TL;DR

This paper explores fractional Sobolev metrics on spaces of smooth and Sobolev-regular curves, proving local well-posedness of geodesic equations and establishing conditions for strong Riemannian metrics, advancing shape analysis tools.

## Contribution

It introduces and analyzes fractional order Sobolev metrics on curve spaces, proving well-posedness and metric strength conditions, extending previous work on invariant metrics.

## Key findings

- Proves local well-posedness of geodesic equations for fractional Sobolev metrics.
- Shows $H^s$-metrics induce strong Riemannian metrics for $s > 3/2$.
- Generalizes analysis of right invariant metrics on diffeomorphism groups.

## Abstract

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\operatorname{Imm}(S^1,\mathbb{R}^d)$ and on its Sobolev completions $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$. We prove local well-posedness of the geodesic equations both on the Banach manifold $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$ and on the Fr\'{e}chet-manifold $\operatorname{Imm}(S^1,\mathbb{R}^d)$ provided the order of the metric is greater or equal to one. In addition we show that the $H^s$-metric induces a strong Riemannian metric on the Banach manifold $\mathcal{I}^{s}(S^1,\mathbb{R}^{d})$ of the same order $s$, provided $s>\frac 32$. These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.