From the diffeology article Differentiable manifolds also generalize smoothness They are normally defined as topological manifolds with an atlas whose transition maps are smooth which is used to pull back the differential structure emphasis mine The point is not whether differentiable manifolds can be defined without an atlas I already pointed out that they can also be defined with sheaves but that the classical definition that most mathematicians are still using is the atlas definition Markus is correct that the choice of an atlas for a topological manifold doesn t matter but to be a manifold I will contend that it must have some atlas Defining the sphere as a set of zeroes is implicitly embedding the sphere into another space It has already been pointed out several times that a topological sphere does not have a unique differentiable manifold structure How is the differentiable manifold structure induced by the embedding and more to the point how can one understand the corresponding differentiable manifold structure on a topological sphere without mentioning this embedding Topological spaces are not equipped with embeddings a priori That a sphere with a differentiable manifold structure has a topological embedding into euclidean space does not at all imply that the differentiable structure was induced by this embedding