Could someone please help me with: if N,M are diffeomorphic manifolds, what property do they share that non-diffeomorphic manifolds do not share?. I have thought that if A,B were non-diffeomorphic (with dimA=dimB=n), certain functions (i.e, with their respective coord. representations) from A into IR^n that are differentiable , would not be differentiable, in the same sense as with A, as functions from B into IR^n. Any other one?. Thanks.