Fortunate,

You are quite correct. GR doesn't need an embedding space to describe curvature, and in general, with machinery of differential geometry, one doesn't need embedding spaces. However, they can be useful and even insightful to picture something. That doesn't mean the embedding space is real in any physical sense, but it's just a way of seeing things.

And I hope you'll agree that it gives us a pretty slick way to visualize deSitter space-time.

And finally, you're also correct that one shouldn't say space-time is expanding. Give me a big red "-1" there. I get those from to time (but everybody does -- beware those who don't realize how easy it is to get those -1s or worse with this stuff!) Change that to "space-time of an expanding universe".

-Richard