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 $V \cap S$:
- $X$ is differentiable.
- $X$ a homeomorphism in terms of the neighborhoods.
- $\forall q \in U$, 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.
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 $V \cap S$:
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'.