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Math Help - Coordinate Charts and inverse of functions.

  1. #1
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    Coordinate Charts and inverse of functions.

    Hello,

    i have a question about this situation. Let denote T^2=S^1 \times S^1 the Torus.
    And let f:\mathbb{R}^2 \rightarrow T^2 be the projection defined by
    f(x,y)=(e^{i2\pi x},e^{i2\pi y}).

    The claim is now, if \phi is a chart for \mathbb{R}^2 , s.t. f_{|dom\phi} is injective, then \phi \circ (f_{|dom\phi})^{-1} is a chart for T^2.

    i don't understand why the composition is a chart?

    Can you help me?

    Regards
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  2. #2
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    As a chart means it's a 1-1 diffeomorphism in the domain that defined the map. Let V=dom(\phi), U=f(V), W=\phi^{-1}(V)
    Since f is injective in V, f is a 1-1 diffeomorphism between V and f(V).
    And \phi is defined to be a 1-1 diffeomorphism between W and \phi(W)=V.
    Compose two 1-1 diffeomorphism we get a 1-1 diffeomorphism.
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