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I Simpler way to prove smoothness

  1. Jan 31, 2017 #1
    So if ##M, N## are smooth manifolds then ##F: M \rightarrow N## is smooth if given ##p \in M##, there is a smooth chart ##(U, \phi)## containing ##p## and a smooth chart ##(V, \psi)## containing ##F(p)## such that ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \mathbb{R}^n## is smooth.

    If I wanted to prove that a given function was smooth, are there any faster ways other than showing that its coordinate representation is smooth? For example, I just did a question where I had to show that ##T(M \times N)## is diffeomorphic to ##T(M) \times T(N)##. I had to explicitly construct a bijection between the two manifolds then show that the coordinate representations of ##F## and ##F^{-1}## were smooth. This was a big pain. I wish there was a theorem I could have appealed to instead.
     
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  3. Feb 1, 2017 #2

    micromass

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    Inverse function theorem?
    Sections of the tangent bundle?

    It really depends on what you have seen already.
     
  4. Feb 2, 2017 #3
    Thanks, micromass. I haven't learned the inverse function theorem on manifolds yet, but I suppose it's the usual inverse function theorem applied to the coordinate representation of the map I'm interested in. I am still early in my study of smooth manifolds - I'll be more patient.
     
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