Help with smooth manifolds

1. Dec 4, 2003

franznietzsche

i am trying to solve this problem:

Give the paraboloid $$y_{3}=(y_{1})^2+(y_{2})^2$$ the structure of a smooth manifold.

But i am unsure what it means by structure. Can anyone give me some help here?

Last edited: Dec 5, 2003
2. Dec 5, 2003

franznietzsche

i kept trying to solve it on my own, but all i was able to get was:

$$x^1(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{2})^2}$$

$$x^2(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{1})^2}$$

$$x^3(y_{1},y_{2},y_{3})=(y_{1})^2 + (y_{2})^2$$

is that the answer? i'm feeling very lost here.

edit:

Ok i think i figured it out...

i set the local coordinates $$x^1,x^2$$ to lie in the$$y^1;y^3,y^2;y^3$$ planes respectively. thus from the above I get:

$$x^1(y_{1},0,y_{3})=\pm\sqrt{y_{3}}$$

$$x^2(0,y_{2},y_{3})=\pm\sqrt{y_{3}}$$

I also realised that the manifold is 2-dimensional (being embedded in $$E_{3}$$) so there is no $$x^3$$ coordinate.

If any one can give any confirmation on this answer i would greatly appreciate it.

Last edited: Dec 5, 2003