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 indipendent from ZFC (or are conjectured to be)?