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Notes on the Nash embedding theorem

11 May, 2016 in expository, math.AP, math.DG, math.MG | Tags: , , | by | 29 comments

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold {M} can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space {{\bf R}^d}. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let {M} be a compact manifold. Then there exists an embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In fact, if {M} is {n}-dimensional, one can take {d} to equal {2n}, which is often best possible (easy examples include the circle {{\bf R}/{\bf Z}} which embeds into {{\bf R}^2} but not {{\bf R}^1}, or the Klein bottle that embeds into {{\bf R}^4} but not {{\bf R}^3}). One can also relax the compactness hypothesis on {M} to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take {d} equal to {2n+1}.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let {(M,g)} be a compact Riemannian manifold. Then there exists a isometric embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In order to obtain the isometric embedding, the dimension {d} has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

\displaystyle d = \max( n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)

 

is attained, which I believe is still the record for large {n}. (In the converse direction, one cannot do better than {d = \frac{n(n+1)}{2}}, basically because this is the number of degrees of freedom in the Riemannian metric {g}.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of {d}, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in {d} is that it allows one to freely exploit the very useful tool of pairing together two maps {u_1: M \rightarrow {\bf R}^{d_1}}, {u_2: M \rightarrow {\bf R}^{d_2}} to form a combined map {(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}} that can be closer to an embedding or an isometric embedding than the original maps {u_1,u_2}. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of {M} and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

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