# K\"ahler structures on spaces of framed curves

## TL;DR

This paper characterizes the space of framed loops in three-dimensional space as an infinite-dimensional K"ahler manifold, describes its geometric properties, and relates it to polygon spaces and planar loops, with implications for shape analysis.

## Contribution

It identifies the framed loop space with a complex Grassmannian, explicitly describes geodesics, and analyzes curvature and symmetries, extending previous work on polygon and planar loop spaces.

## Key findings

- Framed loop space is an infinite-dimensional K"ahler manifold.
- The space of immersed loops is a symplectic reduction of the framed loop space.
- The space and its quotient are nonnegatively curved.

## Abstract

We consider the space $\mathcal{M}$ of Euclidean similarity classes of framed loops in $\mathbb{R}^3$. Framed loop space is shown to be an infinite-dimensional K\"{a}hler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of $\mathcal{M}$ by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of $\mathcal{M}$ explicitly. Using this description, we show that $\mathcal{M}$ and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in $\mathcal{M}$ as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on $\mathcal{M}$ is shown to be Hamiltonian and this is used to characterize the critical points of the weighted total twist functional.