*Preprint*

**Inserted:** 5 jan 2005

**Last Updated:** 13 feb 2011

**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 $c:S^1\rightarrow R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\rightarrow R^n$ 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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