# Unified Definition of a Surface

As I noted in this question, there's a lot of inconsistent terminology in use with regard to "curves", "smooth curves" etc and similar comments could apply to the definition of a surface. To get all of this organized in my brain, I would like to define a curve as a topological manifold of dimension $1$ and a smooth curve as a smooth manifold of dimension $1$. Similarly, I would like to define a surface as a topological $2$-manifold and a smooth surface as a smooth $2$-manifold. To me, this approach feels intuitive and seems to be somewhat standard; for example, I believe this is the approach that Lee takes in his Smooth Manifolds text.

Montiel and Ros in their Curves and Surfaces text define a "smooth surface" this way:

We will say that a non-empty set $S \subset \mathbb{R}^3$ is a (smooth) surface if, for each $p \in S$, there exist an open set $U \subset \mathbb{R}^2$, an open neighbourhood $V$ of $p$ in $S$, and a differentiable map $X : U \rightarrow \mathbb{R}^3$ such that the following hold.

i: $X(U) = V$

ii: $X : U \rightarrow V$ is a homeomorphism.

iii: $dX_q : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is injective for all $q \in U$.

It seems what the authors are in fact defining here is a $2$-manifold $S$ that is smoothly immersive at each of its points. Is this the right way to interpret their definition with respect to the terminology described above using manifolds? Is there more succinct/accurate terminology that can be used to describe the situation?

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It sounds like a perfectly reasonable definition of an embedded submanifold to me. Being an embedded submanifold is a stronger condition than being an immersed submanifold: for example, there is a topology and smooth structure on $S^1 \subset \mathbb{R}^2$ making it immersed but not embedded. (Take the obvious immersion $[0, 1) \to S^1$.) – Zhen Lin Jan 31 '12 at 22:32