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Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally)

What if it is smooth?

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    $\begingroup$ Poznyak proved that every two-dimensional Riemannian manifold locally isometrically embeds into $\mathbb{R}^4$. Globally, you need $\mathbb{R}^{10}$. $\endgroup$
    – Ben McKay
    Feb 18, 2016 at 12:39
  • $\begingroup$ Lots of obvious examples: all surfaces in $\mathbb{R}^3$, and all products of plane curves, so lots of flat tori. $\endgroup$
    – Ben McKay
    Feb 18, 2016 at 12:42
  • $\begingroup$ Can you list some counterexamples that can't embedd into R^4 globally thank you. $\endgroup$ Feb 18, 2016 at 12:43

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I found the following in the notes to chapter 2 of the book of Han and Hong: "Isometric embedding of Riemannian manifolds in Euclidean spaces" which is a great source for questions about isometric immersions and embeddings.

The real projective plane $\mathbb{RP}^2$ (with the standard (constant Gauss curvature 1) metric from $S^2$) has no $C^2$ isometric embedding in $\mathbb{R}^4$ (in particular this rules out smooth embeddings as in your question). Han and Hong cite the 1970 survey "Embeddings and immersions in Riemannian geometry" by Gromov and Rokhlin, in particular see Appendix 4.

This result deserves a little more comment. Note that there do exist embeddings of $\mathbb{RP}^2$ into $\mathbb{R}^4$, see e.g. wikipedia. What Gromov and Rokhlin prove is that there are none with strictly positive curvature.

Gromov and Rokhlin state that it is open whether isometric embeddings for $\mathbb{RP}^2$ in $\mathbb{R}^4$ exist for $C^r$ with $1<r<2$ (recall that for $r\leq1$, Nash-Kuiper applies).

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This is more of a remark than an answer and addresses the weaker condition of asking for an immersion rather than an embedding.

I assume that you want the immersion to be smooth (or at least $C^2$) since, otherwise, by Nash-Kuiper, there is no obstruction.

On the other hand, according to Gromov's Partial Differential Relations (published in 1986, so it may be out of date on this), "There is no single known obstruction for isometric $C^\infty$-immersions of surfaces into Riemannian $4$-manifolds". (See 3.2.4, pp. 289-294.)

More specifically, the last time I checked, it was unknown whether $\mathbb{RP}^2$ endowed with any metric of positive curvature could be isometrically immersed into $\mathbb{R}^4$ and it was still unknown whether the hyperbolic plane (with Gauss curvature $-1$) could be isometrically immersed in $\mathbb{R^4}$. The latter can be isometrically embedded in $\mathbb{R}^5$, as can $\mathbb{RP}^2$ with a metric of constant positive Gauss curvature. See Gromov's PDR for details.

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