ON THE GEOMETRY OF STAR-SHAPED CURVES IN Rⁿ

Abstract

The manifold M of star-shaped curves in Rⁿ is considered via the theory of connections on vector bundles, and cyclic D-modules. The appropriate notion of an ‘integral curve' (i.e. certain admissible deformations) on 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 (Korteweg-de Vries) hierarchy.