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I've understood it such that diffeomorphisms are the best notion of equality between manifolds (basically a diffeomorphism is just a smooth homeomorphism), but what exactly are the properties that are preserved after a diffeomorphic map between manifolds (say M and N)?

If I would guess it's a diffeomorphism from M to N preserves all the properties that a homeomorphism preserves PLUS the differentiability of M (say it's C^k). But are there any more important properties to keep in mind?