Stability of \(B\)-incompleteness
type: Item
instance of: scholarly article
title: Stability of \(B\)-incompleteness
zbMATH Open document ID: 0792.53062
DOI: 10.1016/0034-4877(92)90018-V
published in: Reports on Mathematical Physics
publication date: 1994-07-24T00:00:00Z
review text: The author defines a natural topology on the space of all linear, torsion free, connections. He uses the fact that the difference of any two such connections is a symmetric \({1 \choose 2}\)-tensor field and defines a basis of neighbourhoods of a connection \(\Gamma\) by taking \[ {\mathcal E}(\Omega,\Gamma) = \{\Gamma'| \{\Gamma'(x) - \Gamma(x)| x \in M\} \subset \Omega\}, \] where \(\{\Omega\}\) is a basis of neighbourhoods of the zero section in the \(({1\atop 2})\)-tensor bundle. This induces a topology on the space of Lorentz metrics (For any Lorentz metric \(g\) let \(\Gamma^ g\) be the associated Levi-Civita connection. A typical neighbourhood of \(g\) is of the form \(\{g'\mid \Gamma^{g'} \in {\mathcal E}(\Omega,\Gamma^ g)\})\). He then considers a curve \(\gamma: [t_ 0,T) \to M\) which either has two endpoints or has the property that for all \(p \in M\) there exists a neighbourhood \(U\) and a \(t_ 1 \in [t_ 0,T)\) such that \(U\cap \gamma([t_ 1,T)) = \emptyset\). He shows that for any Lorentz metric \(g\) there exists a neighborhood of \(g\) such that either \(\gamma\) is \(b\)-bounded with respect to all metrics in this neighbourhood or \(\gamma\) is not \(b\)-bounded with respect to any such metric. He does not give an example why his assumption on \(\gamma\) should be necessary.
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 83C99
zbMATH DE Number: 459125
zbMATH Keywords: \(b\)-boundary
zbMATH Keywords: generalized affine parameter
zbMATH Keywords: topologies of metrics
zbMATH Keywords: topologies of connections
zbMATH Keywords: causality violation
zbMATH Keywords: linear, torsion free, connections
zbMATH Keywords: Lorentz metrics
author: Andrés Viña
reviewed by: Marcus Kriele
MaRDI profile type: Publication
cites work: The Large Scale Structure of Space-Time
cites work: A new definition of singular points in general relativity
cites work: The space of Lorentz metrics
cites work: Q5825483
cites work: Spinor Structure of Space-Times in General Relativity. I
cites work: Q4153990
full work available at URL: 0034-4877(92)90018-v
OpenAlex ID: W2048082866
description: scientific article; zbMATH DE number 459125