wikipedia has a list of some statements known to be independent of ZFC, but there are no examples in differential geometry/differential topology; does anyone know if there is one?
or if maybe there's a way of showing something along the lines of "smooth manifolds are very well-behaved, so there can't be any"
According to Are Perfectly Normal Manifolds Metrisable?, the question whether every perfectly normal manifold is metrisable, is independent of ZFC.