Results 1 to 3 of 3

Thread: submanifolds

  1. #1
    Member
    Joined
    Oct 2010
    Posts
    131

    submanifolds

    Hello,

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

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

    $\displaystyle
    g(x)=(x-\frac{e^x-e^{-x}}{e^x+e^{-x}}, \frac{2}{e^x+e^{-x}})
    $

    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
    $\displaystyle
    C^{\infty} $ surface? or only a $\displaystyle C^k$. How can i decide about this question.
    I hope you can help me in my problem.

    Regards
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,793
    Thanks
    3035
    g(x)= (x- tanh(x), sech(x))

    Are those infinitely differentiable for all x?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2010
    Posts
    131
    Yes i think so, they are infinitely differentiable for all x.

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

    Regards
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Submanifolds
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Feb 10th 2010, 08:54 PM

Search Tags


/mathhelpforum @mathhelpforum