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- W4291216687 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." @default.
- W4291216687 created "2022-08-13" @default.
- W4291216687 creator A5073878869 @default.
- W4291216687 date "2017-01-11" @default.
- W4291216687 modified "2023-09-29" @default.
- W4291216687 title "Kahler structures on spaces of framed curves" @default.
- W4291216687 doi "https://doi.org/10.48550/arxiv.1701.03183" @default.
- W4291216687 hasPublicationYear "2017" @default.
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