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Smooth maps between manifolds domain restriction

  1. Jan 23, 2014 #1
    Let ##M## and ##N## be smooth manifolds and let ##F:M \to N## be a smooth map. Iff ##(U,\phi)## is a chart on ##M## and ##(V,\psi)## is a chart on ##N## then the coordinate representation of ##F## is given by ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \to \psi(V)##. My question is: why one restricts the domain of ##\psi \circ F \circ \phi^{-1}## to ##\phi(U \cap F^{-1}(V))## and not just ##\phi(U)##? I see one can run the risk that ##F(U) \cap V = \varnothing## and that ##\psi(\varnothing)## is not well defined. Is this the reason for the restriction on the domain?
     
  2. jcsd
  3. Jan 24, 2014 #2
    I think what you're trying to say is right.

    The basic idea is that your chart homeomorphism ##\psi## is defined only on ##V##. Thus, we have to start with elements of ##\phi(U)## (the domain of ##\phi^{-1}##) that have images that ##F## can map into ##V##. Otherwise, we could have a point ##u\in U## such that ##F(u)\not\in V##, so ##\psi(u)## is undefined.
     
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