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Math Help - submanifolds

  1. #1
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    submanifolds

    Hello,

    i have a question to this exercise:
    Show that the surface of revolution defined by this curve is a submanifold:

    g: \mathbb{R}_{>0}->\mathbb{R}^2,

     <br />
g(x)=(x-\frac{e^x-e^{-x}}{e^x+e^{-x}}, \frac{2}{e^x+e^{-x}})<br />

    I could show, that this surface is a regular 2-dimensional submanifold.
    Because the derivative of the parametrisation of the surface has at any point dimension 2.
    But now i ask myself what about the smoothness of my submanifold? Is it a
     <br />
C^{\infty} surface? or only a C^k. How can i decide about this question.
    I hope you can help me in my problem.

    Regards
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  2. #2
    MHF Contributor

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    g(x)= (x- tanh(x), sech(x))

    Are those infinitely differentiable for all x?
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  3. #3
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    Yes i think so, they are infinitely differentiable for all x.

    But this is only the "generating" curve. Why must be the surface also C^\infty, if the generating curve is?

    Regards
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