1. ## 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

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

Are those infinitely differentiable for all x?

3. 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