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Thread: 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 $\displaystyle T^2=S^1 \times S^1$ the Torus.
    And let $\displaystyle f:\mathbb{R}^2 \rightarrow T^2$ be the projection defined by
    $\displaystyle f(x,y)=(e^{i2\pi x},e^{i2\pi y})$.

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

    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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