Embedding in the $d+1$ Euclidean space I have heard of the Whitney embedding theorem where any $d$-dimensional manifold can be embedded in $\mathbf{R}^{2d}$. My question is whether the necessity of going up to $2d$ rather than $d+1$ is only due to the global properties of the manifold (e.g. avoiding self intersections) or not?
Here is a more precise statement of what I am looking for:
Take a $d$-dimensional manifold $\mathcal{M}$ equipped with a metric $g$. For any point $p\in M$ is it possible to choose a neighborhood $U$ containing $p$, and a map $\phi:U\rightarrow \mathbf{R}^{d+1}$, where $\mathbf{R}^{d+1}$ is equipped with a flat Euclidean metric $\eta$, and we have the pullback $\phi^* \eta=g$ (restricted to $U$ of course)?
 A: The answer depends on the degree of smoothness: In this type of questions you cannot simply say "metric" or "map," you have to specify the degree of differentiability.

*

*On the positive side, if all what you care about are $C^1$-smooth maps, then indeed, every continuous metric on an $n$-dimensional $C^1$-smooth manifold can be $C^1$-smoothly locally isometrically embedded in the Euclidean space $E^{n+1}$ (this is a part of the Nash-Kuiper isometric embedding theorem).


*On the negative side, there are $C^\infty$-smooth metrics on surfaces which do not admit $C^3$-smooth local isometric embeddings in the Euclidean 3-space $E^3$, see for instance:
Nikolai  Nadirashvili, Yu Yuan, Counterexamples for Local Isometric Embedding, arXiv:math/0208127, 2002
and also examples of $C^{2,1}$-metrics without local $C^3$-isometric embeddings:
Nadirashvili, Nikolai; Yuan, Yu, Improving Pogorelov’s isometric embedding counterexample, Calc. Var. Partial Differ. Equ. 32, No. 3, 319-323 (2008). ZBL1147.53017.
A: This is not an answer to your question about metrics, but as far as global obstructions, go, avoiding self-intersections is not the only one. There's also a classic topological obstruction even to having an immersion of a smooth manifold $M$ into $\mathbb{R}^d$ (which is everywhere locally an embedding but may not be injective), which is that the sum of the tangent bundle and the normal bundle must be the pullback of the tangent bundle of $\mathbb{R}^d$, which is to say, the trivial bundle of rank $d$. (The Hirsch-Smale theorem implies that if $\dim M < d$ this necessary condition is in fact sufficient to guarantee the existence of an immersion.)
This generally imposes nontrivial conditions on the characteristic classes of $M$; classically it's known (see the Hirsch-Smale link above) that we can find a certain product of real projective spaces of dimension $n$ whose Stiefel-Whitney classes imply that it can't be immersed into $\mathbb{R}^{2n - \alpha(n) - 1}$ where $\alpha(n)$ is the number of $1$s in the binary expansion of $n$, and on the other hand Cohen showed that every compact smooth $n$-manifold immerses into $\mathbb{R}^{2n - \alpha(n)}$ so these counterexamples are tight.
A special case, which is a nice exercise, is to take $n = 2^k$ a power of $2$, where we get that $\mathbb{RP}^{2^k}$ does not immerse into $\mathbb{R}^{2^{k+1} - 2}$.
