Preprint
Inserted: 5 jan 2005
Last Updated: 11 feb 2022
Journal: arXiv
Year: 2004
Abstract:
In this paper we study geometries on the manifold of curves.
We define a manifold $M$ where objects $c\in M$ are curves, which we
parameterize as $c:S^1\to \real^n$ ($n\ge 2$, $S^1$ is the circle). Given a
curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all
deformations $h:S^1\to\real^n$ of $c$.
We discuss Riemannian and Finsler metrics $F(c,h)$ on this manifold $M$, and
in particular the case of the geometric $H^0$ metric $F(c,h)=\int
h
^2ds$ of
normal deformations $h$ of $c$; we study the existence of minimal geodesics of
$H^0$ under constraints; we moreover propose a conformal version of the $H^0$
metric.
Keywords: space of curves
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