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