Total curvature of curves in Riemannian manifolds
type: Item
instance of: scholarly article
title: Total curvature of curves in Riemannian manifolds
zbMATH Open document ID: 1188.53054
author: V. Fernández Mateos
author: Marco Castrillón López
author: Jaime Muñoz Masqué
published in: Differential Geometry and its Applications
publication date: 2010-03-29T00:00:00Z
review text: The global properties of the curvature of curves embedded in \({\mathbb{R}}^2\) or in \({\mathbb{R}}^3\) have been studied in many papers. In the present paper, these properties are studied for curves embedded in arbitrary Riemannian manifolds. The integral curvature of a curve is approached using geodesic polygons inscribed in the curve. For non-positive sectional curvature of the manifold, the integral curvature of a curve is the least upper bound of the total curvatures of geodesic polygons inscribed in the curve. Examples are given such that this property fails for positive sectional curvature. For oriented surfaces, the curvature can be equipped with the sign and the above result holds for the signed total curvature. At the end, it is proved that the least upper bound for signed total curvature among simple inscribed polygons for curves closing a simple domain on surfaces of positive curvature is necessarily \(2\pi\).
reviewed by: Zdeněk Dušek
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5687509
zbMATH Keywords: geodesic polygon
zbMATH Keywords: total curvature
zbMATH Keywords: Riemannian manifold
MaRDI profile type: Publication
full work available at URL: j.difgeo.2009.10.008
OpenAlex ID: W2636359
Wikidata QID: Q115357157
cites work: Q4692973
cites work: A geometric inequality for the total curvature of plane curves
cites work: Global inequalities for curves and surfaces in three-space
cites work: Total curvature and packing of knots
cites work: Q3499993
cites work: Total curvature, ropelength and crossing number of thick knots
cites work: Q4003026
cites work: TOTAL CURVATURES OF HOLONOMIC LINKS
cites work: The shape of a figure-eight under the curve shortening flow
cites work: Total curvature of graphs in space
cites work: Q5731354
cites work: On the total curvature of knots
cites work: Q4328659
cites work: Q3499992
DOI: 10.1016/J.DIFGEO.2009.10.008
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description: scientific article; zbMATH DE number 5687509