Geometric invariants of non-smooth framed curves

TL;DR

This paper compares two framing methods for non-smooth curves, derives their geometric invariants, and discusses their applications in variational problems, enhancing understanding of curve geometry in non-smooth contexts.

Contribution

It introduces a comparison between the Serret-Frenet frame and the RPAF for $W^{2,2}$-curves, deriving invariants and exploring applications.

Findings

Derived curvature and torsion for RPAF

Compared geometric invariants of different frames

Discussed applications in variational problems

Abstract

We compare the Serret-Frenet frame with a {\em relatively parallel adapted frame} (RPAF) introduced by Bishop to parametrize $W^{2,2}$-curves. Next, we derive the geometric invariants, curvature and torsion, with the RPAF associated to the curve. Finally, we discuss applications of the two approaches in variational problems.