Horizontal lifts of spacelike curves with non-differentiable endpoints
type: Item
instance of: scholarly article
title: Horizontal lifts of spacelike curves with non-differentiable endpoints
zbMATH Open document ID: 0611.53053
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1987-01-01T00:00:00Z
review text: Given a differentiable principal fibre bundle P over a differentiable manifold M and a compact continuous curve \(\sigma\) : [0,L]\(\to M\) which is differentiable on the half-open interval [0,L), when is it possible to find a horizontal lift of \(\sigma\) into P over the entire interval [0,L]? It is not difficult to show that this can be done if \(\sigma\) has finite length in some (hence, any) Riemannian metric on M. The situation is more complicated if we look at Lorentz metrics on M for which \(\sigma\) is a spacelike curve of finite length. It is shown here that if, in such a metric, \(\sigma\), when parametrized by arc-length, has timelike acceleration, then the same conclusion can be drawn: A horizontal lift of \(\sigma\) into P over [0,L) extends continuously to L. Counter-examples can be found if the acceleration is not timelike. One application is given, with P the tangent bundle over a b-complete Lorentz manifold M: If \(\sigma\) : [0,L)\(\to M\) is a unit-speed spacelike curve with \(\nabla _{{\dot \sigma}}{\dot \sigma}=\kappa N\) for some unit timelike vector N, if L is finite and \(| \nabla _{{\dot \sigma}}N|\) is bounded, then \(\sigma\) is differentiably extendible to and past L.
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 3988189
zbMATH Keywords: principal fibre bundle
zbMATH Keywords: horizontal lift
zbMATH Keywords: Lorentz metrics
zbMATH Keywords: timelike acceleration
zbMATH Keywords: b-complete Lorentz manifold
zbMATH Keywords: spacelike curve
author: Steven (Stacey) G. Harris
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228321
OpenAlex ID: W2002621344
cites work: Q3913406
cites work: Closed and complete spacelike hypersurfaces in Minkowski space
cites work: The Large Scale Structure of Space-Time
cites work: Q5731354
DOI: 10.2748/TMJ/1178228321
description: scientific article; zbMATH DE number 3988189