Vtyjkyuk

What geometric object $(x,y,z,w)$ is defined by the equation $xw-yzle 0$ in $mathbb R^4$? Using the parameticals, can we describe this manifold? Prove some diffeomorphism between this geometrical object and others?

The set $D$ is not a manifold; in fact, it's not even a manifold with boundary. It has nonempty interior (the set of all points where $xw-yz<0$), so if it were going to be a manifold with boundary it would have to be $4$-dimensional, and thus its boundary would have to be a $3$-manifold, which it's not. The interior, however, is a $4$-manifold, because it's an open subset of $mathbb R^4$.

Here's how to see that $partial D$ is not a $3$-manifold. It's the set where $xw = yz$, and I claim this set is actually a cone on a torus. By that I mean it is homeomorphic to the quotient of the space $mathbb S^1timesmathbb S^1 times [0,infty)$, with the entire set $mathbb S^1times mathbb S^1 times 0$ collapsed to a point. Consider the map $Fcolon mathbb S^1timesmathbb S^1 times [0,infty) to mathbb R^4$, given by
beginalign*
$$
(x,y,z,w) =

F(theta,phi,r) = (rcosphi + r costheta, r sintheta+ rsinphi,

rsintheta- rsinphi, rcosphi- rcostheta),
$$
where $theta$ and $phi$ are interpreted mod $2pi$, so they represent points on a circle. A straightforward computation shows that $F$ maps into $partial D$ and collapses the entire set $mathbb S^1times mathbb S^1 times 0$ to the origin. A little more work shows that its image is all of $partial D$ and it doesn't make any other identifications, so it descends to the quotient to show that $partial D$ is homeomorphic to the cone I mentioned above.

This cone is not a $3$-manifold. The proof that it's not is a little technical, but here goes. Every point in a $3$-manifold has a neighborhood $U$ homeomorphic to a $3$-ball, and therefore when the given point $p$ is removed from $U$, the punctured set $Usmallsetminus p$ is simply connected. But if $U$ is any neighborhood of the origin in $partial D$, when you remove the origin you get a set homeomorphic to an open subset of $mathbb S^1times mathbb S^1 times (0,infty)$, and containing a torus of the form $mathbb S^1times mathbb S^1 times varepsilon$ for some small $varepsilon$. The set $Usmallsetminus 0$ retracts onto this torus. Since a retract of a simply connected space is simply connected, while the torus is not, this shows that no punctured neighborhood of the origin in $partial D$ is simply connected. Thus $partial D$ is not a $3$-manifold, and $D$ is not a manifold with boundary.