Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications

TL;DR

This paper presents a method for constructing simple, piecewise geodesic interpolations of Jordan curves on Riemannian manifolds, with applications to generalized Green's theorem and signature uniqueness.

Contribution

It introduces an explicit construction method for geodesic interpolations of Jordan curves, allowing for arbitrary fineness and inclusion of specified points.

Findings

Constructed geodesic interpolations can be arbitrarily fine.

Applied to prove generalized Green's theorem.

Established uniqueness of signature for certain planar curves.

Abstract

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be included in our construction. Then we present two applications of our main results: the generalized Green's theorem and the uniqueness of signature for planar Jordan curves with finite p-variation for 1<=p<2.