I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#metadata_info_tab_contents). Let me recall the setup so I can make my question as accurate as possible:

We have a (possibly non-compact) Riemannian manifold $(M,g)$ and a short immersion $z:M\rightarrow \mathbb{R}^k$ wich induces a new metric $h$ on $M$. Shortness means that the tensor $\delta=g-h$ is again possitive definite.

We also have an open cover $N_p$ along with a subordinate partition of unity $\phi_p$ such that every neighborhood $N_p$ meets only finite others. So we now focus on the immersion of $N_p$ into $\mathbb{R}^k$.

Now Nash wants to approximate the tensor $\frac{1}{2}\delta_{ij}$ by another positive tensor $\beta_{ij}$. He does this by first finding a set $M_{\mu,\nu}$ of p.d.s.m (possitive definite symmetric matrices) such that for any p.d.s.m. $A$ we have $A=\sum_{\mu,\nu}C^{*}_{\mu,\nu}M_{\mu,\nu}$ where the coefficients $C^*_{\mu,\nu}$ depend smoothly on $A$.

Here is what is bugging me. Nash writes: enter image description here

So if I understand correctly, since $\beta_{ij}$ is a smooth choice of a p.d.s.m for any point $x\in N_p$, we can view $\beta_{ij}$ as smooth immersion of $N_p$ into the space of p.d.s.m's , which is an open cone in the vector space of symmetric matrices. My question is: Why can't we choose $\beta_{ij}=\delta_{ij}$?

Again, on top of page $391$ it is stated that we can make $\beta_{ij}$ arbitrarily close to $\delta_{ij}$. But what stops us from having them exactly equal? Any expository notes on this theorem will also be greatly appreciated!

  • $\begingroup$ I read several years ago an outreaching article about smooth fractals appearing in this theorem, and it was mentioned that infinitely many "corrugations" are needed to ensure the isometric embedding. So maybe adding more and more corrugations could correspond to the two tensors getting closer and closer to each other. $\endgroup$ – Sylvain JULIEN Apr 7 '20 at 13:40
  • $\begingroup$ @SylvainJULIEN I don't think so. $\delta$ (or more precisely $\frac{1}{2}\delta$ captures the info of how to make the "corrugations" (in Nash's paper we have spirals since we work in codim $2$ and not corrugations that work in codim $1$). The corugations are ment to improve the induced metric $h$ and how much improvement is needed is measured by $\delta$. At least that is my understanding. $\endgroup$ – Nick A. Apr 7 '20 at 13:46
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    $\begingroup$ Otherwise, maybe this will help you: math.ucla.edu/~hmkhang24/expository/nash.pdf $\endgroup$ – Sylvain JULIEN Apr 7 '20 at 14:01
  • $\begingroup$ Those are notes on Nash's smooth isometric embedding theorem, which uses entirely different techniques. For his $C^1$ theorem, see arxiv.org/abs/1606.02551, and for some additional comments also ihes.fr/~gromov/wp-content/uploads/2018/08/nash-copy-Oct9.pdf and ams.org/journals/bull/2017-54-02/S0273-0979-2016-01560-6 $\endgroup$ – slcvtq Apr 9 '20 at 8:50

I believe the only issue is that Nash only assumes $g$ to be continuous. So the error tensor is also only continuous, and with the procedure as you suggest it, the functions $a_\nu$ (following Nash's notation) would only be continuous. This would be a problem since $a_\nu$ are one of the building blocks of the "Nash twist" (13), so that the next 'immersion' in the sequence would only be continuous - and hence not even an actual immersion, not something that induces a new metric with a new associated error. The procedure would break down.

Of course, the paper is already extremely interesting if you assume everything to be smooth other than the very final $C^1$ 'isometric immersion' limit of the approximating immersions, so I think this is an ignorable technicality for anyone reading the paper for the first time.

But it is an interesting technicality, since it is roughly of the same nature as the 'loss of derivatives' problem which motivates Nash's famous introduction of smoothing operators into the Newton method, in his subsequent paper on smooth isometric embeddings.

Tangentially, it's worth noting that Gromov generalized this part of Nash's paper in a very appealing way. From page 170 of his "Partial Differential Relations": let $A\subset\mathbb{R}^q$ be a connected embedded $C^\ell$-submanifold, let $V$ be a compact manifold, and let $f_0$ be a $C^r$ map from $V$ into the interior of the convex hull of $A$, with $r\leq\ell$. Then $f_0$ can be written a finite constant convex combination of $C^r$ maps from $V$ into $A$.

In Nash's context, $\mathbb{R}^q$ is something like the vector space of symmetric matrices while $A$ is the submanifold of rank-1 matrices. I find Gromov's statement quite remarkable, since even in simple contexts (e.g. $q=3$, $A$ a sphere) it seems extremely non-intuitive. Strangely, neither Nash's statement (5) nor its proof seem so non-intuitive.

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    $\begingroup$ Aha! I remember that when I was reading the introduction of the paper I saw that $g$ is assumed only continuous but quickly glossed over this. I should have been more carefull, thank you very much! $\endgroup$ – Nick A. Apr 9 '20 at 8:50

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