This question is inspired by an old UT prelim exam (question 2). Let me abbreviate the question here, getting right to the bonus which is all I need help with.
Let $S^2 \subset \mathbb{R}^3$ be the standard unit sphere, and consider $S^2 \times S^2 \subset \mathbb{R}^3 \times \mathbb{R}^3$. Let $f:S^2 \times S^2 \to \mathbb{R}$ be the square of the Euclidean distance; $f(v,w)=d(v,w)^2$. Set $S=f^{-1}(1/2)$, so that $S$ is a compact $3$-dimensional submanifold of $S^2 \times S^2$. Can you identify $S$?
I think that projection (onto the first factor, say) induces a fiber bundle $p:S \to S^2$, and the preimage of a point is a circle, so I'm looking at an $S^1$-bundle over $S^2$ and I just need to know which one. From searching around I think something called the first Chern class would nail down which one, but it seems to me that some other invariant like (co)homology would work too.
Any thoughts or strategies are appreciated :)
By working directly, one can see that $S$ is the sphere bundle of the tangent bundle of $\mathbb S^2$.
Let $x\in \mathbb S^2$. Then $(x, y) \in S$ if and only if $|x-y| = \frac{1}{\sqrt 2}$. All such $y$ (for each fixed $x$) corresponds to a circle on $\mathbb S^2$, and this circle lies in the plane perpendicular to the vector $x$. The total space of all these plane is obviously the tangent bundle of $\mathbb S^2$ and $S$ are those tangent vectors with length $\frac{1}{2\sqrt 2}$.