finding an inclusion map to show submanifold

Hi there

I've got the following exercise to solve and I don't believe my ideas are that bad but it doesn't seem to be enough yet:

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With an injective curve that does not cross the axis z and a non-zero derivative values consider the surface parametrised by the rotation of around the axis z.

Now show that is a submanifold of with dimension 2. State an atlas.

++ (sorry when translating isnt perfect)

Concerning the lecture when then is a parametrisation of with and am I right?

My idea was taking the inverse of this p:

is injective and surely surejective to the image of itself so invertable. So is well-defined. But why is z never 0 ???

Now I want to argument that i is an inclusion map that embeds into . Is this the right way? Is this what one calls an "inclusion map"?

Then I'll have to show that the jacobian of i is injective everywhere on . This is equivalent to showing that the jacobian is invertible for all points on the surface is this correct?

What do you suggest?

Regards

Huberscher