Prove that x^4+y^4=1 is a manifold

[andromeda2]

Can someone help me out whit this
I proved for the circle, but I can't prove it for this

-Prove that x^4+y^4=1 (the set of points) is a manifold

For the circle it was easy, but how do I take on this case?

Thanks

 

[wofsy]

how have you tried to prove it?

 

[andromeda2]

I was thinking about a kind of stereographic projection
but I didn't know how to do this

 

[wofsy]

a kind of stereographic projection to create coordinate neighborhoods?

 

[andromeda2]

yes exactly
Do you think that this will work? Then I try it again

 

[wofsy]

my guess is that you should be able to identify open intervals that project back to give coordinate charts. This would give you a differentiable manifold. You then need to argue that it is diffeo to a circle.

 

[gel]

Let f(x,y) be any smooth real valued function, and a be a real number. Then, the set of points (x,y) for which f(x,y) = a and df/dx, df/dy are not both zero is a manifold.

In this case, f(x,y) = x^4 + y^4, and df/dx = 4x^3, df/dy=4y^3, which can't both be zero when f=1. So, yes, the locus of points you describe is a manifold.

That's a very general method. In this case you can show that the locus of points is diffeomorphic to the circle, as already mentioned.

 

[wofsy]

Let f(x,y) be any smooth real valued function, and a be a real number. Then, the set of points (x,y) for which f(x,y) = a and df/dx, df/dy are not both zero is a manifold.

In this case, f(x,y) = x^4 + y^4, and df/dx = 4x^3, df/dy=4y^3, which can't both be zero when f=1. So, yes, the locus of points you describe is a manifold.

That's a very general method. In this case you can show that the locus of points is diffeomorphic to the circle, as already mentioned.

right but the problem was to construct coordinate neighborhoods. An appeal to the implicit function theorem is not a direct construction. But there is more to it. Once you know it is a manifold you need to prove that it is a topological circle (without appealing to the structure theorem for 1 manifolds.) With coordinate charts this should be easy to do.

 

[gel]

if A = {(x,y) : x^4+y^4=1} and S={(x,y):x^2+y^2=1} is the unit circle, then you can define

$$\begin{align*} &f\colon A\to S,\ f(x,y) \equiv (\lambda x, \lambda y),\ \lambda = (x^2+y^2)^{-\frac{1}{2}}\\ &g\colon S\to A,\ g(x,y) \equiv (\lambda x, \lambda y),\ \lambda = (x^4+y^4)^{-\frac{1}{4}} \end{align*}$$

which are smooth maps and inverse to each other, showing that A,S are diffeomorphic.