Metrics with prescribed horizontal bundle on spaces of curve

TL;DR

This paper characterizes a broad class of reparametrization invariant metrics on the space of curves that induce a specific prescribed splitting of the tangent bundle, extending previous results on normal-tangential decompositions.

Contribution

It generalizes the characterization of metrics inducing a normal-tangential splitting to those inducing any prescribed tangent bundle splitting.

Findings

Identifies all metrics inducing a given tangent bundle splitting.

Extends previous work on normal-tangential splitting.

Provides a framework for designing metrics with desired geometric properties.

Abstract

We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics $G$ on the space $\operatorname{Imm}(S^1,\mathbb R^2)$ of parametrized regular curves. For many metrics the tangent space $T_c\operatorname{Imm}(S^1,\mathbb R^2)$ at each curve $c$ splits into vertical and horizontal components (with respect to the projection onto the shape space $B_i(S^1,\mathbb R^2)=\operatorname{Imm}(S^1,\mathbb R^2)/\operatorname{Diff}(S^1)$ of unparametrized curves and with respect to the metric $G$). In a previous article we characterized all metrics $G$ such that the induced splitting coincides with the natural splitting into normal and tangential parts. In these notes we extend this analysis to characterize all metrics that induce any prescribed splitting of the tangent bundle.