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What are some specific results in (pseudo)riemannian geometry that require either the axiom of countable choice or the axiom of (countable) dependent choice (DC)?

For instance, does the definition of distance between two points in a (pseudo)riemannian manifold require countable choice or DC when using the infimum of the lengths of infinite curves between the two points?

bonif
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    Why do you think the axiom of choice has anything to do with these? – Asaf Karagila Sep 21 '17 at 01:35
  • I've seen several answers about the AC and its weaker forms and while there seems to be consensus that the full AC is in general not needed the weaker forms are, but this was in the general context of analysis,not of (semi)Riemannian,or differential geometry. I was just curious where in the latter areas that could be the case, if not in the definition of the objects maybe in some results. For instance calculus of variations in manifolds seem to intuitively involve some kind of simultaneous choice. From your commnet one could infer that there is no such example and that this should be obvious? – bonif Sep 21 '17 at 08:58
  • @AsafKaragila, I would appreciate some clarification, – bonif Sep 21 '17 at 13:12
  • @bonif: What clarification do you need? Write down the definition of a Riemann integral in full detail (that's all what is needed to define the Riemannian distance) and check if it requires DC. – Moishe Kohan Sep 22 '17 at 04:33
  • @MoisheCohen Right it is actually trivial to see that with the Riemannian distance no choice is required. But in the semirieammnain case the integral has an additional element, it includes a plus/minus sign choice, because paths can be timelike or spacelike depending on it and this choice must be kept(the sequence of curves to pick the infimum from must consist of either timelike or spacelike curves) , wouldn't this require dependent choice? – bonif Sep 22 '17 at 10:05
  • @bonif: Just write down the detailed definition and you will see that it does not need DC. – Moishe Kohan Sep 22 '17 at 13:12
  • @MoisheCohen The definition of distance certainly doesn't require any choice in the semiriemannian case but my question was much broader. It only took me a quick look at a couple of books on semiriemannian geometry to see that even thouh in my specific example in the OP isn't indeed actually the full AC is used in certain global results concerning the total action of lagrangian densities and well-posedness of the intial value problem in relativity. – bonif Sep 23 '17 at 08:53
  • @AsafKaragila FYI I found results needing full AC in Wald and Hawking&Ellis. Also see https://math.stackexchange.com/questions/10102/where-do-we-need-the-axiom-of-choice-in-riemannian-geometry where the Arzelá-Ascoli theorem is mentioned in relation to geodesics. – bonif Sep 23 '17 at 12:07
  • Hey man, I'm not familiar with almost anything geometric, let alone Riemannian geometry. I'm here because of the axiom of choice. – Asaf Karagila Sep 23 '17 at 12:13
  • And I'm glad to learn anything from Mr. Choice ;-) – bonif Sep 23 '17 at 14:22

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