In globally hyperbolic spacetimes, the locus of multiple maximizing geodesics is locally contractible and homotopy equivalent to the causal future minus the Lorentzian Aubry set.
Lecture Notes on Lorentzian Geometry
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On the locus of multiple maximizing geodesics on a globally hyperbolic spacetime
In globally hyperbolic spacetimes, the locus of multiple maximizing geodesics is locally contractible and homotopy equivalent to the causal future minus the Lorentzian Aubry set.