finding an inclusion map to show submanifold
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:
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?