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There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the literature).

Question: For $n \in \mathbb{N}$ what is the minimal integer $r$ (depending on $n$) such that every point in every $n$-dimensional Riemannian manifold $M$ has an open neighborhood which admits a smooth (resp. analytic) locally closed isometric embedding into $\mathbb{R}^r$?

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The Nash-Kuiper theorem says that for $C^1$ isometric embeddings, $r$ just needs to be $n+1$, so in what follows smooth means $C^2$ or better.

These results are all in chapter 1 of Han and Hong's book "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces," which is a great source for these sorts of results.

Let $s_n=\binom{n+1}{2}=\frac{n(n+1)}{2}$.

Theorem 1.1.6 (Janet-Cartan) states:

Any $n$-dimensional analytic Riemannian manifold admits a local analytic isometric embedding in $\mathbb{R}^{s_n}$.

This is proved using the Cauchy-Kowalevski theorem for analytic solutions of PDE in section 1.1.

Note that this is in some sense best possible since $s_n$ is the number of equations in the system $\sum_{k=1}^q\partial_i u^k\partial_j u^k = g_{ij}$ where $g_{ij}$ are the components of the metric on $M^n$ in some local coordinate system and $u:M^n\rightarrow\mathbb{R}^q$ is the embedding.

Theorem 1.2.4 (Gromov and Rokhlin, Greene) implies:

Any smooth $n$-dimensional Riemannian manifold admits a smooth local isometric embedding in $\mathbb{R}^{s_n+n}$.

It seems that it is an open question whether $r$ can be reduced to $s_n$ for smooth local isometric embeddings. In the notes to chapter 1 of their book, Han and Hong state that this is open even for $n=2$ and cite a 1982 list of problems by Yau.

Note that for some $n$, there are better results; e.g. when $n=2$, $r$ may be taken to be 4.

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    $\begingroup$ A couple of quick comments: 1) The Janet-Cartan theorem is indeed the best possible because it is possible to construct a Riemannian metric for which there is not even a formal solution to the isometric embedding equation at a point. 2) As for extending the Janet-Cartan theorem to smooth metrics, the problem is that the system of PDEs is a really nasty one, unless suitable assumptions are made. Even when $n=2$ there is no complete solution. In higher dimensions there is very little known. I can only speculate that an approach that does not use standard PDE theory has to be used. $\endgroup$ – Deane Yang Oct 30 '17 at 0:21
  • $\begingroup$ Gromov proves a number of theorems about the existence of local isometric embeddings in his book Partial Differential Relations. He also has a recent survey paper (an outgrowth of his talk at the Nash memorial conference) in the Bulletin of the AMS. ams.org.proxy.library.nyu.edu/journals/bull/2017-54-02/… $\endgroup$ – Deane Yang Oct 30 '17 at 0:25
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    $\begingroup$ @DeaneYang Thanks for the comments! Here's a non-proxified link to Gromov's paper: ams.org/journals/bull/2017-54-02/S0273-0979-2016-01551-5 $\endgroup$ – j.c. Oct 30 '17 at 1:18
  • $\begingroup$ Oops. Thanks for posting the correct link. $\endgroup$ – Deane Yang Oct 30 '17 at 1:25

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