If this is your first Greg Egan, just put it down slowly and back away quietly. No one needs to get hurt here. Go find an earlier work as an introduction.
This is hard SF to make Hannu Rajaniemi blush. This is hard SF made of adamantium, or impossibilium. I'm guessing there aren't 200 people in the world who could read this and smile and say, oh, yes, haha, of COURSE it would be like that, wouldn't it?" and maybe ten who could say "ahem, a little error there on page 212 with the rotational forces out of balance with the shear forces, heh heh."
The plot is negligible, and not terribly important. The characters are barely sketched, and the setting less so. There is a Quest, because how else do we get a tour of the world? There's a river that runs uphill, unless you want to go down, in which case look just over there for one that runs the other way.
If I understood it right, which I probably didn't, our heroes are exploring their bowl-shaped part of the world when they find a hole that goes through to the hyperboloid side. The effect of this is intriguing, because it leaves our heroes and the reader equally confused.
The book that this one makes you anticipate is this world's equivalent of Darwin's The Origin of Species. Here, you take the physics and produce a creature that could *possibly* cope in such a setting. Could it evolve from something else? How? Can we imagine that millions of generations of primitive creatures died off except for those that learned not to try to turn around? How did the Siders start out?
I certainly can't fault Egan for not including any of that here.
I didn't enjoy this book, but I'm glad I read it. Someone has to go to the edge of "hey, fellas, what if ..." and by golly, Egan hasn't just gone to the edge here -- he's moved the edge.