Rolled-up dimensions vs surfaces in a higher dimensional space All the accounts of dimensions higher than 4 seem to talk about them being 'rolled up'. Is this different to being confined to a 4D surface that exists in a higher dimensional space?
 A: A "4D surface that exists in a higher dimensional space" is called an embedding in differential geometry. Almost all the work in differential geometry is dedicated to making statements independent of any embedding. Hence, physically, it shouldn't matter if 4D space (i.e. Einstein's description of the gravitational universe) is embedded in a higher dimensional space if we cannot experience the additional dimensions of this embedding.
Likewise, rolled-up dimensions in string theory could be understood by a corresponding embedding, but as long as this does not add any physically measurable extra information, the embedding is unnecessary. Moreover, embeddings will never be unique. You can always add another hypothetical dimension of which the mathematical surface description is independent.
The actual point about rolled-up dimensions is that you can get back to a starting point by moving in one direction only, and especially after an extremely short distance. In 4D space, however, it is not possible to move in all the same direction and finally return to where you started. At least as far as we know today.
A: Yes these are conceptually different. The 'rolled up' dimensions are generally compactified (i.e. very small). Hence, they're not immediately in conflict with observation - we only experience 4 spacetime dimensions of course. For models with branes, typically there are certain boundary conditions that seperate the bulk space from the branes. I should also add that in this case, usually these other dimensions aren't compact.
These two Wikipedia pages give a brief description of the differences: https://en.m.wikipedia.org/wiki/Randall%E2%80%93Sundrum_model https://en.m.wikipedia.org/wiki/Compactification_(physics)
