Gentlemen, I hate to rain on this little higher dimensional parade, but I'm gonna have to. The whole higher point of differential geometry is *you don't need it*. However, it can be a very useful visualizing tool to help you understand.

For example, you can imagine a curved 2D space as a curved surface in a flat 3D space. The defining (and simplest) example is the surface of sphere (that turns out to be constant positive curvature) -- one imagines flat landers on the surface aware of only 2D dimensions. And one extends that up one dimension and imagines a closed 3D space of positive constant curvature that is the "surface volume" of a hypersphere.

One can then work out a mathematical framework about how to do things by using coordinates confined to the surface. The higher flat space is useful because it is Euclidean -- Pythagoras rules -- and you can work out how things work in the curved coordinates from your higher dimensional vantage point because you intuitively know how flat space works.

But then you get to the point. Once you've worked out the machinery for dealing with the curved coordinates that live in the curved space, you realize that you don't need the higher dimensional space at all. That was just a crutch so to speak. And then you realize something more fundamental: what so special about "flat" space in the first place? Once you have the general curved rules to guide you, you realize flat space is just a special case where things work out very simply, or maybe I should work out *as we expect*.

Our minds somehow declare that "real space" must be flat, must be Euclidean, and so if there's any "curvature" funny business going on, it must be because the real space is higher dimensional and we're somehow living on a curved hypersurface embedded therein. But, once you've "got it", you realize that's not the case. There's no reason space must be "flat", and no reason a curved space must be embedded in a higher flat space.

But that's just "space", and by that I mean something with a positive definite metric signature. IOW, the (invariant) notion of "distance" (the norm) is always positive and, in the most simple coordinates involved the Pythagorean sum of squares.

Well, space-time is not positive definite. Even flat space-time, Minkowski, is *non-Euclidean*. It's not positive definite. The invariant "distance"/norm there (s/ds) can be negative. The distance formula involves the differences of squares of the time-like and space-like parts.

So even in flat space-time, we've left our dear Euclid behind and moved on to some abstraction that is not like any "real space" our minds can visualize.

And then GR lets that get curved. But our minds persist in trying to paint pictures, filling in gaps of reality that don't have to be there.

-RIchard