On the existence of focal points near closed geodes\-ics on surfaces
type: Item
instance of: scholarly article
title: On the existence of focal points near closed geodes\-ics on surfaces
zbMATH Open document ID: 1034.53042
DOI: 10.1023/A:1024085309422
author: Marlies Gerber
published in: Geometriae Dedicata
publication date: 2003-08-06T00:00:00Z
review text: Let \(S\) be a compact Riemannian surface. The author introduces the notion of focal points near closed geodesics on \(S\) and gives criteria for the (non)existence of such points. Similarly, analogous definitions for the (non)existence of conjugate points near closed geodesics can be made. The criteria are formulated in terms of the curvature along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. These criteria lead to a new family of examples of surfaces with no focal points. The geodesic flows for these examples are ergodic, but not Anosov. Moreover, the author proves that if \(S\) has no focal points and an inequality relating the curvature of \(S\) to the curvature of the horocycles holds, then the horocycles (considered as curves in \(S\)) are uniformly \(C^{2+\text{Lipschitz}}\).
reviewed by: Ruth Kellerhals
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37D10
zbMATH DE Number: 1960060
zbMATH Keywords: focal points near closed geodesics
zbMATH Keywords: conjugate points near closed geodesics
zbMATH Keywords: curvature
zbMATH Keywords: geodesic flows
MaRDI profile type: Publication
full work available at URL: a:1024085309422
OpenAlex ID: W77280222
description: scientific article; zbMATH DE number 1960060