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Mathematics
Differential Geometry
A claim about smooth maps between smooth manifolds
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[QUOTE="quasar987, post: 6866710, member: 14909"] I think you basically answered yourself there. There is a manifold, and an atlas on it. So that's a collection of local homeomorphisms into euclidean space such that their composition on overlaps are diffeomorphisms. Such a system allows one to define smoothness of a function ##f:M\rightarrow\mathbb{R}## at a point ##p\in M##. Namely, it is "smooth at ##p##" if ##b^{-1}\circ f## is smooth for one, [I]hence for all,[/I] chart (b,V) of the atlas around p. The "hence for all" is due to the compatibility condition that the transition maps ##b\circ a^{-1}## are diffeomorphisms. So now the question our friend is trying to answer says this: "Pick a homeomorphism ##a:U\rightarrow a(U)\subset \mathbb{R}^m##. Is (a,U) compatible with the atlas on M? That is, are [I]all [/I]the transition maps ##b\circ a^{-1}## diffeomorphisms when ##b## comes from the atlas? Show that the answer is yes if and only if a function ##f:U\rightarrow \mathbb{R}## is smooth (relative to the atlas!) iff ##f\circ a^{-1}## is smooth (in the ususal sense for functions on R^m) [/QUOTE]
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Differential Geometry
A claim about smooth maps between smooth manifolds
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