Stability of \(B\)-incompleteness
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.
scientific article; zbMATH DE number 459125