On tori having poles
type: Item
instance of: scholarly article
title: On tori having poles
zbMATH Open document ID: 0607.53030
DOI: 10.1007/BF01388813
author: Nobuhiro Innami
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143345
review text: On a complete Riemannian manifold, a pole is a point which has no conjugate points along any geodesics through it. Given any \(\epsilon\), \(0<\epsilon <1\), there is a nonflat Riemannian metric of total volume 1 on the n-dimensional torus such that the volume \(\lambda\) of the set of all poles satisfies \(1-\epsilon \leq \lambda <1\). On the other hand, a torus is flat if there is a single pole with the additional property that the norm of every Jacobi field along every geodesic emanating from the pole is nondecreasing outside some ball centered at the pole. The proof of the latter statement uses Busemann functions.
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 3982003
zbMATH Keywords: flat torus
zbMATH Keywords: pole
zbMATH Keywords: conjugate points
zbMATH Keywords: Busemann functions
MaRDI profile type: Publication
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cites work: Q5683877
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cites work: Riemannian manifolds without focal points
cites work: An elementary method in the study of nonnegative curvature
cites work: On the fundamental group of compact manifolds of non-positive curvature
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description: scientific article; zbMATH DE number 3982003