Is $\{(x, y, z) \in \mathbb{R}^3: z^2=(x^2 +y^2)^2 \}$ a manifold?

Question

The set in the title can be rewritten as $f^{-1}(0)$, where $f(x,y,z) = z^2 - (x^2 + y^2)^2$, and 0 is not a regular value, so I can't use the implicit function theorem. What techniques can I employ to endow this set with a differential structure? Is it immersed (or embedded) in euclidean space? I would say it isn't embedded, because it is two ellyptic paraboloids, symmetric in relation to the xy plane and intersecting at the origin.

Answer 1

If you take the square root, you have that $z = \pm (x^2 + y^2)$, and you can sketch both of these surfaces simultaneously. It's not that different from the curve $y^2 = x^4$, where you have $y = \pm x^2$. In both cases, you have a local cut point.

Answer 2

To answer the question in the title:

Intuitively, if it is a manifold, it has to be a surface, so locally diffeomorphic to the Euclidean plane. But a neighborhood of $(0,0,0)$ in your set looks like two neighborhoods of the origin $(0,0)$ glued together at the origin.

This is clearly topologically different from a neighborhood of the origin in the Euclidean plane. So it is not a manifold.