An attempt to answer the question "What can we do with Nash's embedding theorem?" is given in the paper http://mathlab.math.scu.edu.tw/mp/pdf/S30N35.pdf Very broad perspective on Nash's imbedding theorem is provided by Gromov's article "Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems" https://www.ams.org/journals/bull/2017-54-02/S0273-0979-2016-01551-5/
One may continue with questions about relations between regularity classes + the topology/geometry of the source manifolds with the dimensions q of the ambient spaces, but the most compelling problems raised by the results of Nash are not about these.
Nash, like Columbus, unwillingly discovered a new land. Refining and improving Nash’s isometric imbedding results would be like building bigger and faster ships than those in which Columbus had crossed the Atlantic.
But what is this new land? What is its geography, geology, ecology? How can one explore and cultivate this land? What can one build on this land? What is its future? It may be hard to decide what this land is but it is easy to say what it is not:
what Nash discovered is not any part of the Riemannian geometry, neither it has much (if anything at all) to do with the classical PDE.
Nash’s theorems are only superficially similar to the existence (and non-existence) results for isometric embeddings that rely on PDE and/or on relations between intrinsic, i.e. induced Riemannian, and extrinsic geometries of submanifolds in Euclidean spaces. (The primary instance of the latter is the proof of the existence of isometric immersions of surfaces with positive curvatures into the Euclidean 3-space $R^3$ by means of elliptic a priori estimates, that are certain bounds on the extrinsic curvature of a locally convex surface $X⊂R^3$ in terms of the intrinsic Gauss curvature of X.)
Nash’s results points in the opposite direction:
typically, the geometry of a Riemannian manifold X has no significant influence on its isometric embeddings to $R^q$.
In order to get an idea of what kind of mathematics may lie in this "opposite direction" we shall look at Nash’s theorems and his proofs from a variety of different perspectives.
