Horizontal lifts of spacelike curves with non-differentiable endpoints
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.
scientific article; zbMATH DE number 3988189