Say we're given a (possibly Pseudo/Semi-) Riemannian manifold $M$. There are two (equivalent) ways to go about analyzing it. The “old” way is consider it as an embedding withing a higher dimensional (possibly Pseudo/Semi-) Euclidean space. There are various theorems about how many dimensions it takes to embed a particular manifold (Nash, Whitney et cetera). Let us take EVERYTHING relevant to be $C^{\infty}$ (the embedding).
Now we can go another way and study $M$ intrinsically. Here we pull out the Riemann and Ricci tensors and the rest of our bag of tricks. This is the standard way of doing business these days.
Since these two approaches are equivalent, then I'm curious how the number of dimensions required for an embedding of a manifold is reflected in the curvature tensors? I find it rather interesting and would happily accept a good reference or prefferably some explanation and a reference for further reading.
NOTE: I'm studying this from a physics point of view, and am looking at how all of the frames in different places on a spacetime $M$ would be related to one another (so I'm interested in the groups over the manifold which related frames globally). This would seem to depend upon the number of dimensions required to embed it (when such frame changes are written as Clifford algebras of the embedding space).
Thank you!