The total absolute curvature of nonclosed curves in \(S^2\). II
type: Item
instance of: scholarly article
title: The total absolute curvature of nonclosed curves in \(S^2\). II
zbMATH Open document ID: 1065.53003
DOI: 10.1007/BF03323378
author: Kazuyuki Enomoto
author: Jin-Ichi Itoh
published in: Results in Mathematics
publication date: 2004-11-26T00:00:00Z
review text: A piecewise \(C^2\) curve in the unit sphere \(S^2\) is called an extremal curve if it has fixed endpoints, end-directions and length, and minimizes the total absolute curvature. In part I [Result. Math. 45, No. 1--2, 21--34 (2004; Zbl 1064.53002)], the authors proved that any extremal curve is a union of geodesic segments and small circular arcs with possible singularities at the endpoints. The present paper is a continuation of the above and the main results can be summarized in the following theorem: Any extremal curve of length less than \({{\pi }\over 2}\) is either a \(C^1\) curve consisting of one small circular arc and at most two geodesic segments, or a geodesic polygon consisting of at most three geodesic segments. If the extremal curve is a geodesic polygon with three sides, then two angles at the vertices must have different signs.
reviewed by: Christos Baikoussis
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 2119242
zbMATH Keywords: curve
zbMATH Keywords: total absolute curvature
MaRDI profile type: Publication
cites work: The total absolute curvature of nonclosed curves in \(S^2\)
full work available at URL: https://doi.org/10.1007/bf03323378
OpenAlex ID: W4238687727
description: scientific article; zbMATH DE number 2119242