On existence and comparison of conjugate points in Riemannian and Lorentzian geometry
type: Item
instance of: scholarly article
title: On existence and comparison of conjugate points in Riemannian and Lorentzian geometry
zbMATH Open document ID: 0585.53037
DOI: 10.1007/BF01450925
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164184
review text: By making use of Jacobi tensors, the existence and comparison of first conjugate points along the geodesics of Riemannian manifolds (or, along the timelike and null geodesics of Lorentzian manifolds) can be studied using the comparison and oscillation theory of linear differential equations. Employing the methods and the results in the comparison and oscillation theory of linear differential equations, a generalized and sharpened Myers theorem, and several generalizations of the Ambrose theorem on existence of conjugate points are obtained. Also, using the comparison theorems for linear differential equations, the first conjugate point along a geodesic of a Riemannian manifold (or, along a timelike or a null geodesic of a Lorentzian manifold) of arbitrary dimension can be compared with the conjugate points along the isotropic geodesics of Riemannian or Lorentzian manifolds (of arbitrary dimensions) by only assuming their Ricci curvatures satisfy certain inequalities. It is shown that, every geodesic (respectively, timelike geodesic) of a 2-dimensional Riemannian manifold (respectively, 2-dimensional Lorentzian manifold) and every null geodesic of a conformally flat Lorentzian manifold of dimension \(n\geq 3\) is isotropic, and therefore, can be used for comparison in the above comparison theorems.
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 3937993
zbMATH Keywords: timelike geodesics
zbMATH Keywords: conjugate points
zbMATH Keywords: oscillation theory of linear differential equations
zbMATH Keywords: Myers theorem
zbMATH Keywords: Ambrose theorem
zbMATH Keywords: comparison theorems
zbMATH Keywords: null geodesic
Wikidata QID: Q115393491
author: Demir N. Kupeli
MaRDI profile type: Publication
cites work: A theorem of Myers
cites work: Q3913406
cites work: Jacobi tensors and Ricci curvature
cites work: A generalization of Myers theorem and an application to relativistic cosmology
cites work: Q3961453
cites work: Q3050744
cites work: Q5584775
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description: scientific article; zbMATH DE number 3937993