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I am currently being confused by two different definitions given in the books "manifolds and differential geometry (by jeffrey lee)" and "differential geometry of curves and surfaces (by do carmo)".

First definition, given by Lee: "A subset $S$ of a smooth n-manifold $M$ is called a "regular submanifold" of dimension $k$ if every point $p \in S $ is in the domain of a chart (U,x) that has the following "regular submanifold property" with respect to $S$: $x(U \cap S ) =x (U) \cap (\mathbb{R}^k \times \{c\})$ for some $c \in \mathbb{R}^{n-k}$.

Note: In Lee's book, a smooth manifold is defined to be a Hausdorff paracompact topological space attached with smoothly compatible charts covering the whole space, i.e. a smooth atlas.

Another (somewhat related) definition given by do Carmo is:

A subset $S \subset \mathbb{R}^3 $ (although I think this can be easily replaced by $\mathbb{R}^n$) is a "regular surface" if, for each $p \in S$, there exists an open $V \subset \mathbb{R}^3$ and a map $x:U \rightarrow V \cap S$ of an open set $U \subset \mathbb{R}^2$ onto $V \cap S \subset \mathbb{R}^3 $ such that

  1. x is $C^\infty$
  2. x is a homeomorphism onto its image
  3. The differential $ dx(q):\mathbb{R}^2 \rightarrow \mathbb{R}^3$ is injective for all $q \in U$.

My question is: How are these definitions related? More precisely,

  1. Given a regular surface in sense of do Carmo, is it always a regular submanifold of dimension 2 of $\mathbb{R}^3$ in sense of Lee? (i.e. is there a 'canonical' way to construct charts having the regular submanifold property?)

  2. Do every regular sub 2-manifold of $\mathbb{R}^3$ in sense of Lee becomes a regular surface in sense of do Carmo?

Thanks.

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  • 1
    $\begingroup$ The two definitions are indeed equivalent. The interesting direction is from do Carmo's definition to Lee's definition. You use the immersion theorem to prove that do Carmo yields an abstract smooth manifold $S$ together with a smooth embedding $S\to M$ (an injective immersion which is a homeomorphism to its image): math.stackexchange.com/questions/1103485/… The question you asked is mostly a duplicate of what is in the link. $\endgroup$ – Moishe Kohan Oct 1 '18 at 1:41
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First of all, you are right about assuming that do Carmo's definition could be easily generalized to submanifolds in any $\mathbb{R}^n$, or in any $M$.
The definitions are slightly different in general. Let's see why.
1) Jeffrey Lee's $\Rightarrow$ do Carmo (generalized).
Suppose $p\in S$, then there is a chart $(U,x)$ in $M$ such that $x(U\cap S)=x(U)\cap (\mathbb{R}^k \times \{0\})$, we can perform a translation in $\mathbb{R}^{n}$ ($n=$ dim $M$) to get $c=0$. Now, just identifying $\mathbb{R}^k$ and $\mathbb{R}^k \times {0}$, we have a parametrization in the sense of do Carmo: $y=x^{-1}|_V: V=x(U\cap S)\rightarrow M$ which is clearly $C^{\infty}$ and a homeomorphism onto its image (because its the restriction of a chart, which is everything good you could desire) and its domain is open in $\mathbb{R}^k$ because $x(U)$ is open in $\mathbb{R}^n$. The injectivity of its differential comes from the fact that $y \circ x|_{U \cap S}=Id_V$, now use the chain rule.

2) do Carmo's (generalized) $\Rightarrow$ Jeffrey Lee's.
Suppose you have $p\in S$, $V\subset M$ open and a map $y:U \rightarrow V\cap S$ satistying 1), 2), 3), with $U$ open subset of $\mathbb{R}^k$. You need to garantee that the map $y^{-1}|:V\cap S \rightarrow U\subset \mathbb{R}^k$ is just the restriction of a chart $(x,W)$ around $p$ in $M$. You can prove that as follows.
Since $y$ is $C^\infty$, one-one and its differential is one-one, it is a diffeomorphism onto its image (one of the implications of the Inverse Function Theorem). Now since this work is all local, and $M$ is locally $\mathbb{R}^n$, we can solve the situation in $\mathbb{R}^n$ and then translate it to $M$ without any difficulty. In these case, once the $y^{-1}$'s are diffeomorphisms from some open sets of $S$ to $\mathbb{R}^k$, you can easily construct extensions for them by just using that $\mathbb{R}^n$ splits orthogonally as $\mathbb{R}^k\times \mathbb{R}^{n-k}$, and use this locally to define a chart in an open subset of $\mathbb{R}^n$ by just sliding up and down along segments orthogonal to your open in $S$, something like $x((p,0)+t(0,v))=y(p,0)+t(0,v)$ (your open may not be contained in $\mathbb{R}^k\times 0$, but I put it like that for the sake of simplicity. I'm sure you can adjust it to work in the "general" case). This would work because in $\mathbb{R}^n$ any diffeomorphism is a chart of its structure.
To conclude, both definitions are equivalent even in the general case when your ambient is any manifold, and your surfaces are actually submanifolds.
Remark: My original answer was wrong in one implication. Full credit to @Thomas, who made me aware of my mistake.

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  • $\begingroup$ The fact that you do not succeed in showing that the map $y^{-1}$ is the restriction of a chart does not mean it's not true. It _is_possible to show this by using the assumptions and the implicit function theorem. For your map $t^3$ the differential is not injective, i.e. 3 is not satisfied. So it's no counterexample. $\endgroup$ – Thomas Oct 1 '18 at 7:14
  • $\begingroup$ I don't use $t^3$ as a parametrization, I use it to put a smooth structrure on the X axis, that will be my S. Then use the inclusion of $\mathbb{R}\rightarrow S$ as a parametrization. $\endgroup$ – Laz Oct 1 '18 at 13:01
  • $\begingroup$ I don't understand how you want to change differentiable structures here. In do Carmo's definition, $x$ is of course $C^\infty$ as a map into the ambient manifold. It is quite easy to find a differentiable map of a neighbourhood of $V$ to $U\times (-c,c)$ which is the inverse of $x$ along $ U\times {0}$, by using a unit normal $n$ along $V$ and looking at $(q,t) \mapsto x(q) + t n(q)$. This uses the linear structure of $\mathbb{R}^3$, but similar constructions can be carried out with some more machinery also if the ambient manifold is a general differentiable manifold. $\endgroup$ – Thomas Oct 1 '18 at 13:55
  • $\begingroup$ ...and if you do not believe me or Moishe Cohen I suggest you take a look at Theorem 3.1 and the pertinent definitions in Morris W. Hirschs "Differential Topology", according to which a subset of a manifold is a ($C^r$) submanifold in the sense of Lee (as above) if and only if it is the image of a manifold under an immersion (of class $C^r$) which is a homeomorphism onto its image - which is just the general form of do Carmos definition. $\endgroup$ – Thomas Oct 1 '18 at 14:27
  • $\begingroup$ It seems that you are right, without looking at any books. Then my argument for the case when the parametrizations are diffeomorphisms onto their images can be extended anyway, because if they are $C^{\infty}$ and with one-one differential we can restrict ourselves to a neighborhood so that it is indeed a diffeo, and use my argument. In what fails my counterexample is in that the differentiable structure induce by the inclusion as parametrization is not the same as the one induce by the cubic map. My sincerest apologies to @J.Doe, to you and everyone in MSE. I will correct my answer ASAP $\endgroup$ – Laz Oct 1 '18 at 18:13

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