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I'm a beginner in Differential Geometry and I'm confused by the terminologies. In do Carmo's Differential Geometry of Curves and Surfaces, a 'regular surface' is a set that is defined locally by a smooth homeomorphism and its differential (page 52 in do Carmo's, ). However, I also read the Elementary Differential Geometry by Pressley, in which the concept of a 'smooth surface' is proposed (page 77 in Pressley's). The definition of a smooth surface is very similar to the regular surface in do Carmo's. I think two definitions in these two books are essentially the same. Am I right?

Definition from do Carmo's: A regular surface is a subset $S$ of $\mathbb{R}^3$ s.t. for each $p \in S$, there exists a neighborhood $V$ of $ S$, open set $U$ of $\mathbb{R}^2$, and a map $X$ from $U$ to $V \cap S$:

  1. $X$ is differentiable.
  2. $X$ a homeomorphism in terms of the neighborhoods.
  3. $\forall q \in U$, the differential of $X$ at $q$ is injective.

Definition from Pressley's: It firstly define what is a surface by defining a subset of $S$ of $\mathbb{R}^3$ with property 2 in do Carmo's. The homeomorphism is called a surface patch. Then comes the definition of regular surface patch as maps that are smooth and satisfy property 3 in do Carmo's. Finally, a smooth surface is defined by: $\forall p \in S$, we can locally find such regular surface patch locally in a neighborhood of p.

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  • $\begingroup$ Probably, yes, but I'd need the definitions. In general, what is a "surface" in your setting depends on what you want to do, and it boils down to the regularity of transition maps between charts. Do you want them to be smooth? You have a smooth surface. $C^1$? You have a $C^1 $ surface. You just want to do topology? Then a chart just needs to be an homeomorphism with $\mathbb{R}^n$. You just want to integrate differential forms on it? Then you can define the dual of differential forms (called "currents") and you can find out that smooth manifolds are a subset of these "generalised" manifolds. $\endgroup$
    – tommy1996q
    Jun 21, 2023 at 16:25
  • $\begingroup$ Anyway, to be 100% sure I'd need the precise definitions. I think that there's only one way to be a smooth manifold, maybe with different definitions, but equivalent $\endgroup$
    – tommy1996q
    Jun 21, 2023 at 16:27
  • $\begingroup$ @tommy1996q Thanks for your response. Since the definitions are very tedious and long so I only gave the page numbers where the definitions are located. Will add those definitions later if I had time. $\endgroup$ Jun 21, 2023 at 16:30
  • $\begingroup$ No problem, but personally I don't have access tot he texts right now, so I can't really help you without them. Again, I bet they're equaivalent, but who knows $\endgroup$
    – tommy1996q
    Jun 21, 2023 at 16:33
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    $\begingroup$ By the way, @tommy1996q, you're way off the deep end, thinking about abstract manifolds. These are all surfaces in $\Bbb R^3$. No chart overlaps. $\endgroup$ Jun 21, 2023 at 16:58

1 Answer 1

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Definition from do Carmo's: A regular surface is a subset $S$ of $\mathbb{R}^3$ s.t. for each $p \in S$, there exists a neighborhood $V$ of $ S$, open set $U$ of $\mathbb{R}^2$, and a map $X$ from $U$ to $V \cap S$:

  1. $X$ is differentiable.
  2. $X$ a homeomorphism in terms of the neighborhoods.
  3. $\forall q \in U$, the differential of $X$ at $q$ is injective.

Definition from Pressley's: It firstly define what is a surface by defining a subset of $S$ of $\mathbb{R}^3$ with property 2 in do Carmo's. The homeomorphism is called a surface patch. Then comes the definition of regular surface patch as maps that are smooth and satisfy property 3 in do Carmo's. Finally, a smooth surface is defined by: $\forall p \in S$, we can locally find such regular surface patch locally in a neighborhood of p.

Latest Edit: Thanks for the answers and instructions given by Ted Shifrin and Tommy1996q. I undeleted and edited this post. The definitions mentioned above are equivalent.

To wit, given a 'smooth surface' $S \subseteq \mathbb{R}^3$, we verify that it is a 'regular surface': Take $p \in S$, we can find a 'regular surface patch', which is a differentiable homeomorphism $X : U \subseteq \mathbb{R}^2 \to \mathbb{R}^3$ such that $p \in X(U)$. Since homeomorphism carries open to open, then this condition immediately gives properties 1 and 2 in do Carmo's. Since $X$ is a 'regular surface patch', its Jacobian has rank 2, which satisfies condition3 in do Carmo's.

Another way around, given a 'regular surface' $S \subseteq \mathbb{R}^3$, $\forall p \in S$, we can find some neighborhoods and a differentiable homeomorphism $X$. Not hard to verify the $X$ is the regular surface patch (More precisely, allowable surface patch). Therefore $S$ is a 'smooth surface'.

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  • $\begingroup$ I typed the definitions correspondingly, in my own language. Hopefully my typings are correct... $\endgroup$ Jun 21, 2023 at 16:48
  • $\begingroup$ @TedShifrin Got it, will delete this one. $\endgroup$ Jun 21, 2023 at 16:49

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