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.
Regularity of optimal transport maps and applica- tions, volume 17 of Tesi
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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.