Let $\mathcal{C}$ be the curve in the $(x, z)$-plane defined by the equation

$$\left(x^{2}-1\right)^{2}+\left(z^{2}-1\right)^{2}=5 .$$

Sketch $\mathcal{C}$, taking care with inflection points.

Let $S$ be the surface of revolution in $\mathbb{R}^{3}$ given by spinning $\mathcal{C}$ about the $z$-axis. Write down an equation defining $S$. Stating any result you use, show that $S$ is a smooth embedded surface.

Let $r$ be the radial coordinate on the $(x, y)$-plane. Show that the Gauss curvature of $S$ vanishes when $r=1$. Are these the only points at which the Gauss curvature of $S$ vanishes? Briefly justify your answer.