How can one best visualize two dimensional manifolds in $\mathbb{R^4}$ (more specifically, $\mathbb{S}^2 \times \mathbb{R})$?

Question

I'm trying to "get a picture", so to speak, of hypersurfaces in $\mathbb{S}^2 \times \mathbb{R}$. One example would be $\left(\dfrac{\cos(u)}{\sqrt{1+u^2}}, \dfrac{\sin(u)}{\sqrt{1+u^2}},\dfrac{u}{\sqrt{1+u^2}}, v \right)$, where $-2\pi \leq u \leq 2\pi$ and $v \in \mathbb{R}$. Now, even though this is a 2-dimensional surface, I haven't found any way to really understand what it looks like. I'm aware this is kind of hard to answer, but any help is appreciated.

Answer 1

Clearly $x^2+y^2+z^2=1$ over your surface, so this is a curve drawn on a $2$-sphere, times a real line. You could call this an infinite curtain.

The $z$-value of the curve keeps increasing, so this is a spring with varying radius$^\star$, spiraling around the $z$-axis monotonically, times a real line.

$^\star$The radius is $\frac{1}{\sqrt{1+4\pi^2}}$ at the extremities, goes monotonically up to $1$ (when $z=0$), then symetrically back to $\frac{1}{\sqrt{1+4\pi^2}}$