Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications

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 \(1\leqslant p<2\).