Rigidity of surfaces with no conjugate points
type: Item
instance of: scholarly article
title: Rigidity of surfaces with no conjugate points
zbMATH Open document ID: 0723.53024
author: Keith Burns
author: Gerhard Knieper
published in: Journal of Differential Geometry
publication date: 1991-01-01T00:00:00Z
review text: E. Hopf proved that any complete Riemannian metric with no conjugate points on the torus \(T^ 2\) is flat. Our paper extends Hopf's argument to obtain sufficient conditions for metrics with no conjugate points on a cylinder or the plane to be flat. We show that a cylinder with no conjugate points, curvature bounded from below whose ends do not open out, must be flat. In the case of the plane, we show that a metric with no conjugate points and curvature bounded from below that satisfies a version of Euclid's parallel axiom must be flat.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 4192560
zbMATH Keywords: rigidity
zbMATH Keywords: Riccati equation
zbMATH Keywords: no conjugate points
zbMATH Keywords: cylinder
zbMATH Keywords: plane
zbMATH Keywords: parallel axiom
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214447537
OpenAlex ID: W1560118477
Wikidata QID: Q115181436
DOI: 10.4310/JDG/1214447537
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description: scientific article; zbMATH DE number 4192560