The fundamental theorem of the local theory for non-smooth curves

TL;DR

This paper generalizes the fundamental theorem of local theory to include non-smooth curves by prescribing curvature and torsion via distributional derivatives, ensuring a unique solution with finite total curvature and torsion.

Contribution

It extends classical smooth curve theory to non-smooth cases using distributional derivatives, broadening the scope of the fundamental theorem.

Findings

Existence of essentially unique non-smooth curve solutions

Solutions have finite total curvature and torsion

Preliminary discussion on continuous data and linear systems

Abstract

We extend the classical fundamental theorem of the local theory of smooth curves to a wider class of non-smooth data. Curvature and torsion are prescribed in terms of the distributional derivative measures of two given functions of bounded variation. The essentially unique non-smooth curve solution has both finite total curvature and total absolute torsion. In case of continuous data, we preliminarly discuss a more general problem involving a linear system of distributional derivative equations.