# Regular Surface and Smooth Surface

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.

• 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. Jun 21, 2023 at 16:25
• 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 Jun 21, 2023 at 16:27
• @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. Jun 21, 2023 at 16:30
• 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 Jun 21, 2023 at 16:33
• 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. Jun 21, 2023 at 16:58

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'.

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