How two determine that whether a function (equation) defines a 1-dimensional manifold

Question

For example, how will you determine that the equation like $x^2 - y^2 + x^4- y^4 = 0$ defines a 1-dimensional manifold or not? I'm new to the manifold theory and I'm pretty confused by its applications.

Any help will be appreciated, thanks!

Answer 1

Here's how to do it by applying the regular level set theorem, also called the pre-image theorem. Write your equation in the form $F(x,y)=0$; in this example, $F(x,y)=x^2-y^2+x^4-y^4$. What you have to do is to verify the hypotheses of the regular level set theorem by proving that for each point $p=(x,y)$, if $p$ satisfies the equation then at least one of the two partial derivatives $\partial F/\partial x |_p$ or $\partial F / \partial y |_p$ is nonzero. Once you've done that, the conclusion of the regular level set theorem says that the solution set of the equation is a 1-manifold.

So for this example, assuming that $x^2-y^2+x^4-y^4=0$, prove that one of $2x+4x^3$ or $-2y-4y^3$ is nonzero.