I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is this statement easy to prove? I mean, does it follow straightly from the locally minimizing property of geodesics in Riemannian manifolds? If yes, please explain it to me, otherwise suggest me some reference where I can found the proof.
It seems that those minimizing/maximizing properties depend upon the geodesics' causal structure (timelike, spacelike or null-like) and, in general, nothing can be said about those geodesic's properties without knowing its causal structure. Is this correct?
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