It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin algebra, the existence of a Suslin line etc. This is not too surprising, since many results in this fields require a nontrivial amount of set theory to be proved. On the other hand, differential geometry seems to be "higher up" in mathematical complexity (i.e., further away from set theoretical questions) and so it seems reasonable to me that no "natural" statement in diff. geom. is independent from ZFC.
Is that the case? That is, are there some statements which are independent from ZFC (or are conjectured to be)?
Edit: many comments (and the only, at the moment, answer) focused on computably undecidable problems. While this does indeed answer the question as formulated (because at least one instance of the problem must be independent), it is not exactly what I had in mind in that it does not offer an explicit example of an independent statement, or at least I don't see how to get one. So a more precise reformulation of my question is (but I am also very interested in more examples of computably indecidable problems): is there an explicit (in some sense of the word) statement in differential geometry that is (or is conjectured to be) independent from ZFC?