Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in general this statement cannot be improved to $C^3$ (there exists a smooth metric on the 2-dimensional unit disk that does not admit a local isometric embedding of class $C^3$ into $\mathbb R^3$). Does anyone know about the $C^2$ case or about a counterexample? I would also be interested in anything between $C^1$ and $C^3$ in terms of Hölder-spaces.
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$\begingroup$ You should link the Nadirashvili-Yuan counterexample. If you mean their 2009 paper, then they don't show exactly what you claim. Namely, the degree of smoothness of $M$ plays an important role. $\endgroup$– Mikhail KatzAug 5, 2014 at 15:30
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$\begingroup$ The paper I mean is arxiv.org/pdf/math.dg/0208127.pdf and dates from 2002. Here the manifold and the metric are smooth and they construct a counterexample or $C^3$-embeddability... $\endgroup$– frogAug 6, 2014 at 9:22
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$\begingroup$ The problem is that paper does not seem to have been published. So I would go by the'08 paper ('08 and not '09 as I wrote earlier). $\endgroup$– Mikhail KatzAug 6, 2014 at 9:27
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$\begingroup$ Thanks, I had a look at that paper and as of 2008 it seems to be open, whether this can be improved to $C^\infty$, and the case $C^2$ seems to be obscure anyway… Do you know any reference on the state of art? $\endgroup$– frogAug 6, 2014 at 9:57
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A torus cannot be embedded isometrically in the $C^2$ case into $R^3$. See this.
Moreover, even the $C^1$ isometric embedding of the torus is pretty convoluted, and IIRC an explicit construction was only given relatively recently. See this mathoverflow question.
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$\begingroup$ Thanks for your answer. I'm aware of the statement about the torus and about its $C^1$-embedding into $\mathbb R^3$ but I'm asking for local embeddings (for global ones it is enough to take a manifold that doesn't admit a topological embedding with codimension one as for example the Klein bottle). $\endgroup$– frogAug 5, 2014 at 13:45