Surfaces Around Closed Principal Curvature Lines, an Inverse Problem

TL;DR

This paper addresses an inverse geometric problem by constructing smooth surfaces containing a specified closed curve as a hyperbolic principal cycle, based on torsion conditions.

Contribution

It introduces a method to construct surfaces with prescribed closed principal curvature lines satisfying torsion constraints.

Findings

Constructed surfaces contain the given curve as a hyperbolic principal cycle.

Established conditions relating torsion of the curve to surface geometry.

Provided a framework for inverse problems in differential geometry.

Abstract

Given a non circular spacial closed curve whose total torsion is an integer multiple of $2\pi$, we construct a germ of a smooth surface that contains it as a hyperbolic principal cycle.