Consider the set in $\mathbb{R}_+^4$, $$ S = \{ (a,b,c,d)\; | \; ad=bc,\; a,b,c,d\geq0\}.$$
What is this manifold? (I.e. does it have a name?)
For starters, I believe $S$ is smooth, connected and compact. It seems to be genus-0 since I can't think of any "holes" in the set.