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23 April 2024

» arxiv » math/0605017

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Sobolev--type metrics in the space of curves
A. C. G. Mennucci ; A. Yezzi ; G. Sundaramoorthi ; math.DG/0312384 ; math.DG/0412454 ; PostScript ; PDF ; Other formats ;
Date:	30 Apr 2006
Subject:	Differential Geometry
Abstract:	We define a manifold $M$ where objects $cin M$ are curves, which we parameterize as $c:S^1 o R^n$ ($nge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1 o R^n$ of $c$. In this paper we study geometries on the manifold of curves, provided by Sobolev--type metrics $H^j$. We study $H^j$ type metrics for the cases $j=1,2$; we prove estimates, and characterize the completion of the space of smooth curves. As a bonus, we prove that the Fr’echet distance of curves (see arXiv:math.DG/0312384) coincides with the distance induced by the ``Finsler $L^infinity$ metric’’ defined in S2.2 in arXiv:math.DG/0412454.
Source:	arXiv, math/0605017
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