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Mathematics
Differential Geometry
A claim about smooth maps between smooth manifolds
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[QUOTE="quasar987, post: 6866181, member: 14909"] That is indeed the question. Working from the hypothesis that the functions ##\psi^1\circ f,\ldots, \psi^n\circ f## are smooth on ##V_p##, can you conclude that ##f## is smooth on ##V_p## ? It is completely tautological. Meaning that just by unraveling the definitions we can see that these are logically equivalent statements. You've already told me that, by definition, ##f## will be smooth on ##V_p## provided that ##F:=\psi\circ f\circ \phi^{-1}## is smooth. This is just a map from an open subset of R^m to R^n so we know what it means for F to be smooth. It means that all of its components ##F^1\,..., F^n## are smooth. What is ##F^j## ? It is ##\psi^j\circ f\circ \phi^{-1}##. Is this smooth? By definition, it means that ##\psi^j\circ f## is smooth. Conveniently, that's our hypothesis. One thing that you may have overlooked and which is possibly making this seem more complicated than it is it that you said This is true, but you can also replace the words "for any pair of charts" by "for SOME pair of chart". That is, if f is smooth relative to some chart, then it is smooth relative to EVERY chart. Prove it if this is new to you. It comes down to the fact that the transition functions ##\psi \circ \phi## are smooth. [/QUOTE]
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Mathematics
Differential Geometry
A claim about smooth maps between smooth manifolds
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