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When I read a book of differential geometry for undergraduate students. A surface in $\mathbb R^3$ is a subject that "locally looks like" a piece of $\mathbb R^2$. So we can have locally parametrization on the surface. But a smooth curve is defined to be a smooth map $$ \alpha: I \to \mathbb R^3, $$ where $I$ is an open interval of $\mathbb R$. Similarly, if we "define" a surface in $\mathbb R^3$ to be a smooth map $$ X:U\to\mathbb R^3, $$ where $U\subset \mathbb R^2$ is a domain, and $X$ is not demanded to be a diffeomorphism. For example, the unit sphere is \begin{align*} X: [0,2\pi)\times[0,\pi)&\to \mathbb R^3,\\ (\theta,\phi)&\mapsto (\cos\theta\sin\phi, \sin\theta\sin\phi,\cos\phi), \end{align*} If we "define" a surface as above, what will happen? I will be appreciated if someone can help.

ZERO
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  • I think you should clarify, what you expect from your definitions. I would say, both of them are not good because, for example, they allow the curve and the surface to have uncontrolled self-intersections. – Sergei Akbarov Mar 22 '17 at 08:46
  • Thanks. For curves, I think there is no problem if it has self-intersections, we still can define curvature and torsion. – ZERO Mar 23 '17 at 02:03

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As you note $X$ is not required to be a diffeomorphism but you do need to require $X$ to be regular, meaning that it is at least $C^1$ and the partial derivatives are linearly independent. Practically speaking if you want to study curvature eventually, you could require $X$ to be $C^2$ for example.

Mikhail Katz
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  • Thank you for your answer. If we require $X$ to be regular and $C^2$, is this "definition" works for most surfaces? It seems not, because the coordinate is now "globally" defined. – ZERO Mar 22 '17 at 08:44
  • The description in terms of regularity is a local description. To get a global description of a manifold $M$ you need to do more work. For example you can require $M$ to be a subset of Euclidean space (of sufficiently high dimension) such that near every point, $M$ is the graph of a vector-valued function over a suitable coordinate plane. – Mikhail Katz Mar 22 '17 at 08:54
  • @ Mikhail Katz Thanks. When I say a surface I mean "a surface in $\mathbb R^3$", it can be immersed (not necessarily embedded). Thus the regularity is just the standard regularity in $\mathbb R^3$, I believe. – ZERO Mar 23 '17 at 02:06