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Notes on the Nash embedding theorem

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 Notes on the Nash embedding theorem 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 Notes on the Nash embedding theorem . 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 Notes on the Nash embedding theorem be a compact manifold. Then there exists an embedding Notes on the Nash embedding theorem from Notes on the Nash embedding theorem to a Euclidean space Notes on the Nash embedding theorem .

In fact, if Notes on the Nash embedding theorem is Notes on the Nash embedding theorem -dimensional, one can take Notes on the Nash embedding theorem to equal Notes on the Nash embedding theorem , which is often best possible (easy examples include the circle Notes on the Nash embedding theorem which embeds into Notes on the Nash embedding theorem but not Notes on the Nash embedding theorem , or the Klein bottle that embeds into Notes on the Nash embedding theorem but not Notes on the Nash embedding theorem ). One can also relax the compactness hypothesis on Notes on the Nash embedding theorem 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 Notes on the Nash embedding theorem equal to Notes on the Nash embedding theorem .

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

Theorem 2 (Nash embedding theorem) Let Notes on the Nash embedding theorem be a compact Riemannian manifold. Then there exists a isometric embedding Notes on the Nash embedding theorem from Notes on the Nash embedding theorem to a Euclidean space Notes on the Nash embedding theorem .

In order to obtain the isometric embedding, the dimension Notes on the Nash embedding theorem has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the boundis attained, which I believe is still the record for large Notes on the Nash embedding theorem . (In the converse direction, one cannot do better than Notes on the Nash embedding theorem , basically because this is the number of degrees of freedom in the Riemannian metric Notes on the Nash embedding theorem .) 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 Notes on the Nash embedding theorem , but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in Notes on the Nash embedding theorem is that it allows one to freely exploit the very useful tool of pairing together two maps Notes on the Nash embedding theorem , Notes on the Nash embedding theorem to form a combined map Notes on the Nash embedding theorem that can be closer to an embedding or an isometric embedding than the original maps Notes on the Nash embedding theorem . This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of Notes on the Nash embedding theorem 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.

— 1. The Whitney embedding theorem —

To prove the Whitney embedding theorem, we first prove a weaker version in which the embedding is replaced by an immersion :

Theorem 3 (Weak Whitney embedding theorem) Let Notes on the Nash embedding theorem be a compact manifold. Then there exists an immersion Notes on the Nash embedding theorem from Notes on the Nash embedding theorem to a Euclidean space Notes on the Nash embedding theorem .

Proof: Our objective is to construct a map Notes on the Nash embedding theorem such that the derivatives Notes on the Nash embedding theorem are linearly independent in Notes on the Nash embedding theorem for each Notes on the Nash embedding theorem . For any given point Notes on the Nash embedding theorem , we have a coordinate chart Notes on the Nash embedding theorem from some neighbourhood Notes on the Nash embedding theorem of Notes on the Nash embedding theorem to Notes on the Nash embedding theorem . If we set Notes on the Nash embedding theorem to be Notes on the Nash embedding theorem multiplied by a suitable cutoff function supported near Notes on the Nash embedding theorem , we see that Notes on the Nash embedding theorem is an immersion in a neighbourhood of Notes on the Nash embedding theorem . Pairing together finitely many of the Notes on the Nash embedding theorem and using compactness, we obtain the claim. Notes on the Nash embedding theorem

Now we upgrade the immersion Notes on the Nash embedding theorem from the above theorem to an embedding by further use of pairing. Let Notes on the Nash embedding theorem be an immersion and Notes on the Nash embedding theorem be points in Notes on the Nash embedding theorem . If Notes on the Nash embedding theorem , then Notes on the Nash embedding theorem is injective in a neighbourhood of Notes on the Nash embedding theorem . If instead Notes on the Nash embedding theorem , then it is possible that Notes on the Nash embedding theorem , but by pairing Notes on the Nash embedding theorem with some scalar function Notes on the Nash embedding theorem that separates Notes on the Nash embedding theorem and Notes on the Nash embedding theorem , we can replace Notes on the Nash embedding theorem by another immersion (in one higher dimension Notes on the Nash embedding theorem ) such that a neighbourhood of Notes on the Nash embedding theorem and a neighbourhood of Notes on the Nash embedding theorem get mapped to disjoint sets. Repeating these procedures finitely many times, using the compactness of Notes on the Nash embedding theorem , we end up with an immersion which is injective, giving the Whitney embedding theorem.

At present, the embedding Notes on the Nash embedding theorem of an Notes on the Nash embedding theorem -dimensional compact manifold Notes on the Nash embedding theorem could be extremely high dimensional. However, if Notes on the Nash embedding theorem , then it is possible to project Notes on the Nash embedding theorem from Notes on the Nash embedding theorem to Notes on the Nash embedding theorem by the random projection trick (discussed inthis previous post). Indeed, if one picks a random element Notes on the Nash embedding theorem of the unit sphere, and then lets Notes on the Nash embedding theorem be the (random) orthogonal projection to the hyperplane Notes on the Nash embedding theorem orthogonal to Notes on the Nash embedding theorem , then it is geometrically obvious that Notes on the Nash embedding theorem will remain an embedding unless Notes on the Nash embedding theorem either is of the form Notes on the Nash embedding theorem for some distinct Notes on the Nash embedding theorem , or lies in the tangent plane to Notes on the Nash embedding theorem at Notes on the Nash embedding theorem for some Notes on the Nash embedding theorem . But the set of all such excluded Notes on the Nash embedding theorem is of dimension at most Notes on the Nash embedding theorem (using, for instance, the Hausdorff notion of dimension), and so for Notes on the Nash embedding theorem almost every Notes on the Nash embedding theorem in Notes on the Nash embedding theorem will avoid this set. Thus one can use these projections to cut the dimension Notes on the Nash embedding theorem down by one for Notes on the Nash embedding theorem ; iterating this observation we can end up with the final value of Notes on the Nash embedding theorem for the Whitney embedding theorem.

Remark 4 The Whitney embedding theorem for Notes on the Nash embedding theorem is more difficult to prove. Using the random projection trick, one can arrive at an immersion Notes on the Nash embedding theorem which is injective except at a finite number of “double points” where Notes on the Nash embedding theorem meets itself transversally (think of projecting a knot in Notes on the Nash embedding theorem randomly down to Notes on the Nash embedding theorem ). One then needs to “push” the double points out of existence using a device known as the “Whitney trick”.

— 2. Reduction to a local isometric embedding theorem —

We now begin the proof of the Nash embedding theorem. In this section we make a series of reductions that reduce the “global” problem of isometric embedding a compact manifold to a “local” problem of turning a near-isometric embedding of a torus into a true isometric embedding.

We first make a convenient (though not absolutely necessary) reduction: in order to prove Theorem, it suffices to do so in the case when Notes on the Nash embedding theorem is a torus Notes on the Nash embedding theorem (equipped with some metric Notes on the Nash embedding theorem which is not necessarily flat). Indeed, if Notes on the Nash embedding theorem is not a torus, we can use the Whitney embedding theorem to embed Notes on the Nash embedding theorem (non-isometrically) into some Euclidean space Notes on the Nash embedding theorem , which by rescaling and then quotienting out by Notes on the Nash embedding theorem lets one assume without loss of generality that Notes on the Nash embedding theorem is some submanifold of a torus Notes on the Nash embedding theorem equipped with some metric Notes on the Nash embedding theorem . One can then use a smooth version of the Tietze extension theorem to extend the metric Notes on the Nash embedding theorem smoothly from Notes on the Nash embedding theorem to all of Notes on the Nash embedding theorem ; this extended metric Notes on the Nash embedding theorem will remain positive definite in some neighbourhood of Notes on the Nash embedding theorem , so by using a suitable (smooth) partition of unity and taking a convex combination of Notes on the Nash embedding theorem with the flat metric on Notes on the Nash embedding theorem , one can find another extension Notes on the Nash embedding theorem of Notes on the Nash embedding theorem to Notes on the Nash embedding theorem that remains positive definite (and symmetric) on all of Notes on the Nash embedding theorem , giving rise to a Riemannian torus Notes on the Nash embedding theorem . Any isometric embedding of this torus into Notes on the Nash embedding theorem will induce an isometric embedding of the original manifold Notes on the Nash embedding theorem , completing the reduction.

The main advantage of this reduction to the torus case is that it gives us a global system of (periodic) coordinates on Notes on the Nash embedding theorem , so that we no longer need to work with local coordinate charts. Also, one can easily use Fourier analysis on the torus to verify the ellipticity properties of the Laplacian that we will need later in the proof. These are however fairly minor conveniences, and it would not be difficult to continue the argument below without having first reduced to the torus case.

Henceforth our manifold Notes on the Nash embedding theorem is assumed to be the torus Notes on the Nash embedding theorem equipped with a Riemannian metric Notes on the Nash embedding theorem , where the indices Notes on the Nash embedding theorem run from Notes on the Nash embedding theorem to Notes on the Nash embedding theorem . Our task is to find an injective map Notes on the Nash embedding theorem which is isometric in the sense that it obeys the system of partial differential equations

Notes on the Nash embedding theorem

for Notes on the Nash embedding theorem , where Notes on the Nash embedding theorem denotes the usual dot product on Notes on the Nash embedding theorem . Let us write this equation as

Notes on the Nash embedding theorem

where Notes on the Nash embedding theorem is the symmetric tensor

Notes on the Nash embedding theorem

The operator Notes on the Nash embedding theorem is a nonlinear differential operator, but it behaves very well with respect to pairing:We can useto obtain a number of very useful reductions (at the cost of worsening the eventual value of Notes on the Nash embedding theorem , which as stated in the introduction we will not be attempting to optimise). First we claim that we can drop the injectivity requirement on Notes on the Nash embedding theorem , that is to say it suffices to show that every Riemannian metric Notes on the Nash embedding theorem on Notes on the Nash embedding theorem is of the form Notes on the Nash embedding theorem for some map Notes on the Nash embedding theorem into some Euclidean space Notes on the Nash embedding theorem . Indeed, suppose that this were the case. Let Notes on the Nash embedding theorem be any (not necessarily isometric) embedding (the existence of which is guaranteed by the Whitney embedding theorem; alternatively, one can use the usual exponential map Notes on the Nash embedding theorem to embed Notes on the Nash embedding theorem into Notes on the Nash embedding theorem ). For Notes on the Nash embedding theorem small enough, the map Notes on the Nash embedding theorem is short in the sense that Notes on the Nash embedding theorem pointwise in the sense of symmetric tensors (or equivalently, the map Notes on the Nash embedding theorem is a contraction from Notes on the Nash embedding theorem to Notes on the Nash embedding theorem ). For such an Notes on the Nash embedding theorem , we can write Notes on the Nash embedding theorem for some Riemannian metric Notes on the Nash embedding theorem . If we then write Notes on the Nash embedding theorem for some (not necessarily injective) map Notes on the Nash embedding theorem , then fromwe see that Notes on the Nash embedding theorem ; since Notes on the Nash embedding theorem inherits its injectivity from the component map Notes on the Nash embedding theorem , this gives the desired isometric embedding.

Call a metric Notes on the Nash embedding theorem on Notes on the Nash embedding theorem good if it is of the form Notes on the Nash embedding theorem for some map Notes on the Nash embedding theorem into a Euclidean space Notes on the Nash embedding theorem . Our task is now to show that every metric is good; the relationtells us that the sum of any two good metrics is good.

In order to make the local theory work later, it will be convenient to introduce the following notion: a map Notes on the Nash embedding theorem is said to be free if, for every point Notes on the Nash embedding theorem , the Notes on the Nash embedding theorem vectors Notes on the Nash embedding theorem , Notes on the Nash embedding theorem and the Notes on the Nash embedding theorem vectors Notes on the Nash embedding theorem , Notes on the Nash embedding theorem are all linearly independent; equivalently, given a further map Notes on the Nash embedding theorem , there are no dependencies whatsoever between the Notes on the Nash embedding theorem scalar functions Notes on the Nash embedding theorem , Notes on the Nash embedding theorem and Notes on the Nash embedding theorem , Notes on the Nash embedding theorem . Clearly, a free map into Notes on the Nash embedding theorem is only possible for Notes on the Nash embedding theorem , and this explains the bulk of the formulaof the best known value of Notes on the Nash embedding theorem .

For any natural number Notes on the Nash embedding theorem , the “Veronese embedding” Notes on the Nash embedding theorem defined by

Notes on the Nash embedding theorem

can easily be verified to be free. From this, one can construct a free map Notes on the Nash embedding theorem by starting with an arbitrary immersion Notes on the Nash embedding theorem and composing it with the Veronese embedding (the fact that the composition is free will follow after several applications of the chain rule).

Given a Riemannian metric Notes on the Nash embedding theorem , one can find a free map Notes on the Nash embedding theorem which is short in the sense that Notes on the Nash embedding theorem , by taking an arbitrary free map and scaling it down by some small scaling factor Notes on the Nash embedding theorem . This gives us a decomposition

Notes on the Nash embedding theorem

for some Riemannian metric Notes on the Nash embedding theorem .

The metric Notes on the Nash embedding theorem is clearly good, so byit would suffice to show that Notes on the Nash embedding theorem is good. What is easy to show is that Notes on the Nash embedding theorem is approximately good :

Proposition 5 Let Notes on the Nash embedding theorem be a Riemannian metric on Notes on the Nash embedding theorem . Then there exists a smooth symmetric tensor Notes on the Nash embedding theorem on Notes on the Nash embedding theorem with the property that Notes on the Nash embedding theorem is good for every Notes on the Nash embedding theorem .

Proof: Roughly speaking, the idea here is to use “tightly wound spirals” to capture various “rank one” components of the metric Notes on the Nash embedding theorem , the point being that if a map Notes on the Nash embedding theorem “oscillates” at some high frequency Notes on the Nash embedding theorem with some “amplitude” Notes on the Nash embedding theorem , then Notes on the Nash embedding theorem is approximately equal to the rank one tensor Notes on the Nash embedding theorem . The argument here is related to the technique of convex integration , which among other things leads to one way to establish the Notes on the Nash embedding theorem of Gromov.

By the spectral theorem, every positive definite tensor Notes on the Nash embedding theorem can be written as a positive linear combination of symmetric rank one tensors Notes on the Nash embedding theorem for some vector Notes on the Nash embedding theorem . By adding some additional rank one tensors if necessary, one can make this decomposition stable, in the sense that any nearby tensor Notes on the Nash embedding theorem is also a positive linear combination of the Notes on the Nash embedding theorem . One can think of Notes on the Nash embedding theorem as the gradient Notes on the Nash embedding theorem of some linear function Notes on the Nash embedding theorem . Using compactness and a smooth partition of unity, one can then arrive at a decomposition

Notes on the Nash embedding theorem

for some finite Notes on the Nash embedding theorem , some smooth scalar functions Notes on the Nash embedding theorem (one can take Notes on the Nash embedding theorem to be linear functions on small coordinate charts, and Notes on the Nash embedding theorem to basically be cutoffs to these charts).

For any Notes on the Nash embedding theorem and Notes on the Nash embedding theorem , consider the “spiral” map Notes on the Nash embedding theorem defined by

Notes on the Nash embedding theorem

Direct computation shows that

Notes on the Nash embedding theorem

Notes on the Nash embedding theorem

and the claim follows by summing in Notes on the Nash embedding theorem (using) and taking Notes on the Nash embedding theorem . Notes on the Nash embedding theorem

The claim then reduces to the following local (perturbative) statement, that shows that the property of being good is stable around a free map:

Theorem 6 (Local embedding) Let Notes on the Nash embedding theorem be a free map. Then Notes on the Nash embedding theorem is good for all symmetric tensors Notes on the Nash embedding theorem sufficiently close to zero in the Notes on the Nash embedding theorem topology.

Indeed, assuming Theorem, and with Notes on the Nash embedding theorem as in Proposition, we have Notes on the Nash embedding theorem good for Notes on the Nash embedding theorem small enough. Byand Proposition, we then have Notes on the Nash embedding theorem good, as required.

The remaining task is to prove Theorem. This is a problem in perturbative PDE, to which we now turn.

— 3. Proof of local embedding —

We are given a free map Notes on the Nash embedding theorem and a small tensor Notes on the Nash embedding theorem . It will suffice to find a perturbation Notes on the Nash embedding theorem of Notes on the Nash embedding theorem that solves the PDE

Notes on the Nash embedding theorem

We can expand the left-hand side and cancel off Notes on the Nash embedding theorem to write this aswhere the symmetric tensor-valued first-order linear operator Notes on the Nash embedding theorem is defined (in terms of the fixed free map Notes on the Nash embedding theorem ) as

Notes on the Nash embedding theorem

To exploit the free nature of Notes on the Nash embedding theorem , we would like to write the operator Notes on the Nash embedding theorem in terms of the inner products Notes on the Nash embedding theorem and Notes on the Nash embedding theorem . After some rearranging using the product rule, we arrive at the representation

Notes on the Nash embedding theorem

Among other things, this allows for a way to right-invert the underdetermined linear operator Notes on the Nash embedding theorem . As Notes on the Nash embedding theorem is free, we can use Cramer’s rule to find smooth maps Notes on the Nash embedding theorem for Notes on the Nash embedding theorem (with Notes on the Nash embedding theorem ) that is dual to Notes on the Nash embedding theorem in the sense that

Notes on the Nash embedding theorem

Notes on the Nash embedding theorem

where Notes on the Nash embedding theorem denotes the Kronecker delta. If one then defines the linear zeroth-order operator Notes on the Nash embedding theorem from symmetric tensors Notes on the Nash embedding theorem to maps Notes on the Nash embedding theorem by the formula

Notes on the Nash embedding theorem

then direct computation shows that Notes on the Nash embedding theorem for any sufficiently regular Notes on the Nash embedding theorem . As a consequence of this, one could try to use the ansatz Notes on the Nash embedding theorem and transform the equationto the fixed point equationOne can hope to solve this equation by standard perturbative techniques, such as the inverse function theorem or the contraction mapping theorem, hopefully exploiting the smallness of Notes on the Nash embedding theorem to obtain the required contraction. Unfortunately we run into a fundamental loss of derivatives problem , in that the quadratic differential operator Notes on the Nash embedding theorem loses a degree of regularity, and this loss is not recovered by the operator Notes on the Nash embedding theorem (which has no smoothing properties).

We know of two ways around this difficulty. The original argument of Nash used what is now known as the Nash-Moser iteration scheme to overcome the loss of derivatives by replacing the simple iterative scheme used in the contraction mapping theorem with a much more rapidly convergent scheme that generalises Newton’s method ; see this previous blog post for a similar idea. The other way out, due to Gunther, is to observe that Notes on the Nash embedding theorem can be factored aswhere Notes on the Nash embedding theorem is a zeroth order quadratic operator Notes on the Nash embedding theorem , so thatcan be written instead as

Notes on the Nash embedding theorem

and using the right-inverse Notes on the Nash embedding theorem , it now suffices to solve the equation(compare with), which can be done perturbatively if Notes on the Nash embedding theorem is indeed zeroth order (e.g. if it is bounded on Hölder spaces such as Notes on the Nash embedding theorem ).

It remains to achieve the desired factoring. We can bilinearise Notes on the Nash embedding theorem as Notes on the Nash embedding theorem , where

Notes on the Nash embedding theorem

The basic point is that when Notes on the Nash embedding theorem is much higher frequency than Notes on the Nash embedding theorem , thenwhich can be approximated by Notes on the Nash embedding theorem applied to some quantity relating to the vector field Notes on the Nash embedding theorem ; similarly if Notes on the Nash embedding theorem is much higher frequency than Notes on the Nash embedding theorem . One can formalise these notions of “much higher frequency” using the machinery of paraproducts , but one can proceed in a slightly more elementary fashion by using the Laplacian operator Notes on the Nash embedding theorem and its (modified) inverse operator Notes on the Nash embedding theorem (which is easily defined on the torus using the Fourier transform, and has good smoothing properties) as a substitute for the paraproduct calculus. We begin by writing

Notes on the Nash embedding theorem

The dangerous term here is Notes on the Nash embedding theorem . Using the product rule and symmetry, we can write

Notes on the Nash embedding theorem

The second term will be “lower order” in that it only involves second derivatives of Notes on the Nash embedding theorem , rather than third derivatives. As for the higher order term Notes on the Nash embedding theorem , the main contribution will come from the terms where Notes on the Nash embedding theorem is higher frequency than Notes on the Nash embedding theorem (since the Laplacian accentuates high frequencies and dampens low frequencies, as can be seen by inspecting the Fourier symbol of the Laplacian). As such, we can profitably use the approximationhere. Indeed, from the product rule we have

Notes on the Nash embedding theorem

Putting all this together, we obtain the decomposition

Notes on the Nash embedding theorem

where

Notes on the Nash embedding theorem

and

Notes on the Nash embedding theorem

If we then use Cramer’s rule to create smooth functions Notes on the Nash embedding theorem dual to the Notes on the Nash embedding theorem in the sense that

Notes on the Nash embedding theorem

Notes on the Nash embedding theorem

then we obtain the desired factorisationwith

Notes on the Nash embedding theorem

Note that Notes on the Nash embedding theorem is the smoothing operator Notes on the Nash embedding theorem applied to quadratic expressions of up to two derivatives of Notes on the Nash embedding theorem . As such, one can show using elliptic ( Schauder ) estimates to show that Notes on the Nash embedding theorem is Lipschitz continuous in the Holder spaces Notes on the Nash embedding theorem for Notes on the Nash embedding theorem (with the Lipschitz constant being small when Notes on the Nash embedding theorem has small norm); this together with the contraction mapping theorem in the Banach space Notes on the Nash embedding theorem is already enough to solve the equationin this space if Notes on the Nash embedding theorem is small enough. This is not quite enough because we also need Notes on the Nash embedding theorem to be smooth; but it is possible (using Schauder estimates and product Hölder estimates) to establish bounds of the form

Notes on the Nash embedding theorem

for any Notes on the Nash embedding theorem (with implied constants depending on Notes on the Nash embedding theorem but independent of Notes on the Nash embedding theorem ), which can be used (for Notes on the Nash embedding theorem small enough) to show that the solution Notes on the Nash embedding theorem constructed by the contraction mapping principle lies in Notes on the Nash embedding theorem for any Notes on the Nash embedding theorem (by showing that the iterates used in the construction remain bounded in these norms), and is thus smooth.

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