Does the set $\{ (x,y,z) | x^2-y^2 = 0, 0 \le y < z \}$ determine a manifold?

Question

Does the set $\{ (x,y,z) | x^2-y^2 = 0, 0 \le y < z \}$ determine a manifold?

From what I understand, the set looks like a triangle above the plane $xy$. However I can't present it in an implicit form.

I think it is a manifold since it doesn't include the $xy$ plane it self. However I don't know how to prove it. I can't think of a proper parameterization.

Any help would be appreciated

Answer 1

Visualizing - take a triangular sheet of paper and fold on an angle bisector. We have a vertex at the origin, a ray heading up from that, and two parts slanting up in the planes $y=x$ and $y=-x$.

Is it a manifold? Well, the most likely place for that to fail is the $z$-axis, and that's a crease. It's not smoothly embedded, but it is a 2-manifold.

Parametrization: $(x,|x|,|x|+t)$ for $x\in(-\infty,\infty)$, $t\in (0,\infty)$.