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). 



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> **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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What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
The [Nash-Kuiper theorem](https://en.wikipedia.org/wiki/Nash_embedding_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:



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



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


