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In "On Einsteins Path" Chapter 11 "Wave Maps in General Relativity", I have seen the following definition for a "regularly embedded manifold $(M,g)$:

Let $q$ be a Euclidean metric on $\mathbb{R}^N$. $(M,g)$ is said to be regularly embedded in $(\mathbb{R}^N,q)$, if it is defined by $N-p$ smooth scalar equations $\Phi^I=0$ on $\mathbb{R}^N$ of rank $p$ on $M$ and if $h$ is the pullback of $q$ under this embedding.

Now when looking online for the definition of a "regularly embedded manifold" I only find search results with the typical definition I already know for a embedded submanifold (or semi-Riemannian submanifold if metrics are involved). Now I am not quite sure, since I don't really see yet why the definitions coincide. From what I know, an embedded semi-Riemannian $d$-dim. manifold $(M,g) \subset (N,h)$ is defined as:

the exist charts on $M$, i.e. $$\varphi: U \subset M \rightarrow V \subset N$$

s.t.
$$ \varphi(U \cap M)= V \cap (\mathbb{R}^p \times \{0\}) $$

and $$g_p= h_p|_{T_p M \times T_pM}$$ is non-degenerate.

Now (using my notations), I can see how I could get $\varphi$ if I am given the $\Phi^I$ and vice versa. But I don't really see how the definitions of the embedded metrics coincide.

User1
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  • IIRC, the two uses of "$p$" are not consistent: In the first description, $p$ is the codimension of $M$, so $M$ has dimension $N - p$. In the second, $M$ is locally modeled on $\mathbf{R}^{p}$, so has dimension $p$. (Also, the second "$M$" in the second description should probably be "$N$"?) <> Be that as it may, if you can see how to pass between the descriptions, can you say specifically what it is you don't understand? Is the missing piece the implicit function theorem? – Andrew D. Hwang Apr 30 '21 at 18:14
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    Thank you, I have edited the mistakes. This always happens because I change the notations I am using in my own notes. I only see that the definitions are equivalent in terms of the manifold, but not how the definitions or requirements for the metric are equivalent, i.e. why restricting the metric to the submanifold (as in the second definition) is the same as the pullback of the metric. – User1 Apr 30 '21 at 18:20
  • Restricting a metric to the image of an immersion is precisely the same as pulling back a metric by that immersion: If each case, we're defining $h_{p}(u, v) = g_{i(p)}(i_{}u, i_{}v)$. :) – Andrew D. Hwang Apr 30 '21 at 18:25
  • The definition in your book is stronger than the standard one (more restrictive); I am ignoring the metrics here. For instance, if you take any non-orientable manifold, it does not admit a regular (in their sense) embedding in $R^N$ for any $N$. Whether the authors of the book realize this or not, I do not know. – Moishe Kohan May 01 '21 at 22:37
  • I now how tried to prove in details why (or why not) the definitions are equivalent. As far as I see, a problem appears when I want to prove the existence of a global function $\Phi$, since the existence of the charts in the second definition only gives local ones. I tried to use your "counterexample" but I do not really see how it works.. – User1 May 02 '21 at 10:21
  • I am not sure how much of the required background you have, but take a look at my answer here. Also here. – Moishe Kohan May 02 '21 at 18:08
  • Yes, thank you, that made it clear! There is no comment in the book where the authors state that this is not the general definition for an embedded submanifold. But I think they are mostly dealing with oriented manifolds. Maybe it is the same in that case? – User1 May 03 '21 at 08:51
  • No, there are oriented examples as well, for instance $CP^2$. – Moishe Kohan May 03 '21 at 12:41

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