Where do we need the axiom of choice in Riemannian geometry? A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the details of Riemannian geometry (or the real analysis it's based on, or the topology, or the theory of vector spaces, etc...) to try and find a theorem or construction that uses the axiom of choice, or one of its logical equivalences. So do you know of any result in Riemannian geometry that needs the axiom of choice? They should be there somewhere, I particularily suspect that one or more is hidden in the basic topology results one uses.

It looks to me like the Arzela\--Ascoli theorem needs at least some weak form of choice. (I have started an MO question to clarify this.) One often uses this in geometry; for example, to guarantee the existence of minimizing geodesics connecting pairs of points.

Edit: See Andres Caicedo's answer on MO (at above link). The answer is affirmative. Also, the database list of equivalents he mentions contains some very innocuous-looking statements that I bet your friend has never thought twice about using.