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

1. Jun 13, 2009

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

Last edited: Jun 14, 2009
2. Jun 14, 2009

wofsy

Re: diffeomorphism

how have you tried to prove it?

3. Jun 14, 2009

andromeda2

Re: diffeomorphism

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

4. Jun 14, 2009

wofsy

Re: diffeomorphism

a kind of stereographic projection to create coordinate neighborhoods?

5. Jun 14, 2009

andromeda2

Re: diffeomorphism

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

6. Jun 14, 2009

wofsy

Re: diffeomorphism

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.

7. Jun 14, 2009

gel

Re: diffeomorphism

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.

8. Jun 14, 2009

wofsy

Re: diffeomorphism

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.

9. Jun 14, 2009

gel

Re: diffeomorphism

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.