How Do You Structure a Paraboloid as a Smooth Manifold?

[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?

 

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


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

 

[M98Ranger]

Sure, I can help you with this problem. To give the paraboloid the structure of a smooth manifold, we need to define a smooth atlas on it. A smooth atlas is a collection of charts that cover the entire manifold and have smooth transition functions between them.

To start, we need to define the topological structure of the paraboloid. Since the paraboloid is a subset of Euclidean space, we can use the standard topology on it. This means that open sets on the paraboloid are defined as the intersection of open sets in Euclidean space with the paraboloid.

Next, we need to define charts on the paraboloid. A chart is a map from an open set on the manifold to an open set in Euclidean space. For the paraboloid, we can use the following two charts:

Chart 1: U = {(y1, y2) | y1 > 0} --> R^2
f(y1, y2) = (y1, y2)

Chart 2: V = {(y1, y2) | y1 < 0} --> R^2
g(y1, y2) = (y1, y2)

Notice that the charts cover the entire paraboloid and their domains overlap on the boundary (y1 = 0).

Finally, we need to define the transition functions between the charts. Since the charts overlap on the boundary, we can use the identity function as the transition function between them. This means that h = g o f^-1 = (y1, y2) on the overlap region.

With these charts and transition functions, we have defined a smooth atlas on the paraboloid. This gives the paraboloid the structure of a smooth manifold. I hope this helps! Let me know if you have any other questions.