On the geometry of star-shaped curves in $\mathbb{R}^n$

Abstract

The manifold $\mathcal{M}$ of star-shaped curves in $\mathbb{R}^n$ is considered via the theory of connections on vector bundles, and cyclic $\mathcal{D}$-modules. The appropriate notion of an "integral curve" (i.e. certain admissible deformations) on $\mathcal{M}$ is defined, and the resulting space of admissible deformations is classified via iso-spectral flows, which are shown to be described by equations from the $n$-KdV hierarchy.