Inserted: 5 jan 2005
Last Updated: 13 feb 2011
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
^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